Business math

Business math mid 2nd Sem

1. Chapter 1:
Introduction
to Business
Mathematics

2. What is Business Mathematics? Importance of Business Mathematics
0 Business mathematics 9/24/2015 12:47:00 AM

What is mathematics?


Mathematics: Mathematics is the science of order, space,
quantity and relation. It is that science in which unknown
magnitudes and relations are derived from known or assumed
ones by use of operations defined or derived from defined
operations.


Types of mathematics: There are two important types of
mathematics


 Abstract mathematics: Mathematics that is used to do the
general operations or activities in our daily life is called abstract
mathematics. For example, accounting, financial mathematics.

3.  Applied mathematics: Mathematics that is pure which
deals with analysis, observation and experiences of various
facts. For example, applied physics, chemistry and medical
science


What is business mathematics?


Business mathematics: Mathematics that is used by
commercial enterprises to record daily transactions forecast
demand and supply as well as other commercial operations
known as business mathematics.




Importance of business mathematics:
Business mathematics is essential to keep track of an
organization day to day operation. The importance of business
mathematics briefly mentioned below


 To record daily transaction of business: An organization
uses business mathematics to keep record of their daily
transaction. How much product they sold today? How much
profit organization earned from today's operation etc. are
calculated using business mathematics.

4.  To forecast production: Organizations analyzes there
product demand and uses business mathematics to determine
how much production should be done meet up these demand.
 To forecast sales volume: Once an organization determine
their production they calculate their sales volume using
business mathematics.
 To calculate profit or loss: Organizations uses business
mathematics to calculate their total cost (TC), total revenue
(TR) and total profit (TP) from the operation.
 To reduce wastage: If a company forecast their production
than they can determine what resources they need such labor,
funds etc. And business mathematics is essential to determine
these resources. This way an organization can reduce wastage
of resources.
you might want to see conceptual framework for financial
reporting


What is prime number?


Prime Number: A number which is not exactly divisible by any
number except itself and unity (1) is called prime number. Such
as, 2,3,5,7,11,17


Example mathematics of prime number

5. Problem: How many prime numbers are between 1 to 100?


Solution: There are 25 prime numbers such as,
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,
83,89,97


Problem: Is the number 97 prime?


Solution: Yes! It is a prime number. An approximate square root
of 97 is 10. The prime less than 10 is 2,3,5,7. 97 is not divisible
by any numbers. So 97 is a prime numbers.


Problem: Is the number 161 prime?


Solution: An approximate square root of 161 is 13. The prime
less than 13 is 2,3,5,7,11. 161 is dvisible by (161/7)= 23. So 161
is not a prime number.

6. What is integer (whole number)?


Integer (whole number): Integer are the whole numbers
either positive, negative or zero.


Positive integer: 1,7,8,16
Negative integer: -5,-7,-8


Zero(0) is an even integer. It is neither positive nor negative.
An integer is said to be even (2n) if it is divisible by 2 otherwise
it is said to be odd (2n+1) or (2n-1)


The formula of consecutive integer is
n, (n+1), (n+2),(n+3)………….Here n is an integer


Example mathematics of integer


Problem: The difference between the square of two
consecutive integer is 47. Find them


Solution: Let the numbers be n and (n+1)

7. According to the question,


(n+1)² - n² = 47
or, n²+2n+1-n² = 47
or, 2n+1 = 47
or, 2n = 46
or, n = 46÷2
or, n = 23


So (n+1) = (23+1) = 24


Therefore the required numbers = 23 and 24


Problem: The difference of the square of two consecutive
integer is 53. Find them


Solution: Let the numbers n and (n+1)


According to the question,


(n+1)² - n² =53
or, n²+2n+1+n² = 53
or, 2n+1 = 53

8. or, 2n = 52
or, n = 52÷2
or, n = 26


So (n+1) = (26+1) = 27


Therefore, the required numbers are 26 and 27

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Simple Definition of Business Mathematics And
Advantages of Business Mathematics
"
Business mathematics may be define different
mathematical formulas or mathematical steps which are taken for
development in business . Large number of business theories
which is used to solve business problems are included in business
mathematics .

"

9. Explanation of Definition of Business Mathematics


From this simple definition , we find that business mathematics is nothing
more than different formulas and theories like interest rate , annuity rate ,
matrix theory , linear programming theory and probability theory and many
more . With these formulas and theories business can calculate many
solutions of different problems .


Advantages of Business Mathematics
 With interest rate of math’s, businessman can calculate the interest on debt
, loan or bonds .
 With matrix , businessman can calculate salary bill of different department
and branches .
 With Linear programming , business can determine the quantity of two
products at which profit is maximize or cost is minimize .
 With discounting and factoring technique businessman can calculate present
value of bill , rat of discounting the bill and banker's gain
 With currency translation formula of math’s, businessman can cost of
currency between two countries .
 With Assignment solution technique businessman can solve transportation
problems.
 Large numbers of business estimations are done on the basis of probability
theory of business mathematics.
 Market research bureau can use consistency data theory of business
mathematics for calculating estimating profit , capital and sale of business .

The Importance of Basic Math in Business
by Carol Deeb, Demand Media
Business ownership requires more than skill in creating a product or talent at
providing a service. Overseeing the finances of your company is key to survival
and success. Understanding basic business math is necessary for profitable

10. operations and accurate record keeping. Knowing how to add, subtract, multiply,
divide, round and use percentages and fractions is the minimum you need to
price your product and meet your budget. If math is not your strength, partner
with someone who can take over that role or hire a trusted employee to help
your operation stay in the black and grow responsibly.
Calculate Production Costs
Before you formally establish your business, you must estimate the cost to
manufacture or acquire your product or perform your service. Adding all expenses
associated with making or buying items helps you realize if you can be
competitive with other companies and profitable enough to sustain your business
and make a reasonable income. In addition to the standard costs of production,
such as materials and machinery, add accompanying expenses, such as shipping,
labor, interest on debt, storage and marketing. The basis to your business plan is
an accurate representation of how much you will spend on each item.
Determine Pricing
To ensure you can operate your business and produce enough cash flow to invest
into your enterprise, you must charge enough for your product to be profitable.
Markup is the difference between your merchandise cost and the selling price,
giving you gross profit. If your operations require a large markup, such as 70
percent, you may not be competitive in your industry if other companies sell the
same items for less. Once you have determined your markup, one way to
calculate the retail price is to divide using percents or decimals. For example, if a
product costs $10 to produce and your markup is 35 percent, subtract .35 from 1
(or 100 percent), which gives you .65, which is 65 percent. To calculate the price
of your product, divide 10 by .65, which rounds to $15.38.

If you want to determine the net profit for a certain time period, you will need to
subtract returns, costs to produce an item and operating expenses from your
total amount of sales, or gross revenue, during that time. Discounts on products,
depreciation on equipment and taxes also must be calculated and subtracted
from revenue. To arrive at your net profit, add any interest you earned from
credit extended to customers, which is reflected as a percent of the amount each
person owes. Your net profit helps you understand if you are charging enough for
your product and selling an adequate volume to continue to operate your
business or even expand.

11. Analyze Finances
To analyze the overall financial health of your business, you will need to project
revenue and expenses for the future. It's important to understand the impact to
your accounting records when you change a number to reflect an increase or
decrease in future sales. Estimating how much an employee affects revenue will
indicate if you can afford to add to your staff and if the profits realized will be
worth the expense. If a competitor starts selling a cheaper product, you may need
to calculate the amount by which your volume must increase if you reduce prices.
You may need to know if you can afford to expand your operations to improve
sales. Using basic business math to understand how these types of actions impact
your overall finances is imperative before taking your business to the next level.

The importance of mathematics
Mathematical thinking is important for all members of a modern society as a habit of
mind for its use in the workplace, business and finance; and for personal decision-
making. Mathematics is fundamental to national prosperity in providing tools for
understanding science, engineering, technology and economics. It is essential in public
decision-making and for participation in the knowledge economy.
Mathematics equips pupils with uniquely powerful ways to describe, analyse and
change the world. It can stimulate moments of pleasure and wonder for all pupils when
they solve a problem for the first time, discover a more elegant solution, or notice
hidden connections. Pupils who are functional in mathematics and financially capable
are able to think independently in applied and abstract ways, and can reason, solve
problems and assess risk.
Mathematics is a creative discipline. The language of mathematics is international. The
subject transcends cultural boundaries and its importance is universally recognised.
Mathematics has developed over time as a means of solving problems and also for its
own sake.

Scope and importance of business mathematics?
Best Answer: Mathematics is an important subject and knowledge of it enhances a
person's reasoning, problem-solving skills, and in general, the ability to think. Hence it is
important for understanding almost every subject whether science and technology,
medicine, the economy, or business and finance. Mathematical tools such as the theory
of chaos are used to mapping market trends and forecasting of the same. Statistics and
probability which are branches of mathematics are used in everyday business and

12. economics. Mathematics also form an important part of accounting, and many
accountancy companies prefer graduates with joint degrees with mathematics rather
than just an accountancy qualification. Financial Mathematics and Business
Mathematics form two important branches of mathematics in today's world and these
are direct application of mathematics to business and economics. Examples of applied
maths such as probability theory and management science, such as queuing theory,
time-series analysis, linear programming all are vital maths for business.

13. Chapter 2:
Number
System

14. History of Numbers
Numbers and counting have become an integral part of our everyday life,
especially when we take into account the modern computer.These words you
are reading have been recorded on a computer using a code of one s and
zeros. It is an interesting story how these digits have come to dominate our
world.
Numbers Around the World
Presently, the earliest known archaeological evidence of any form of writing
or counting are scratch marks on a bone from 150,000 years ago. But the
first really solid evidence of counting,in the form of the number one, is from
a mere twenty-thousand years ago. An ishango bone was found in the
Congo with two identical markings of sixty scratches each and equally
numbered groups on the back.These markings are a certain indication of
counting and they mark a defining moment in western civilization.
1

Zoologists tell us that mammals other than humans are only able to count
up to three or four, while our early ancestors were able to count
further.They believed that the necessity for numbers became more apparent
when humans started to build their own houses, as opposed to living in
caves and the like.
Anthropologists tell us that in Suma, in about 4,000 BCE, Sumerians used
tokens to represent numbers, an improvement over notches in a stick or
bone. A very important development from using tokens to represent
numbers was that in addition to adding tokens you can also take away,
giving birth to arithmetic, an event of major significance.The Sumerian’s
tokens made possible the arithmetic required for them to assess wealth,
calculate profit and loss and even more importantly, to collect taxes, as well
as keep permanent records. The standard belief is that in this way numbers
became the world’s first writings and thus accounting was born.
More primitive societies, such as the Wiligree of Central Australia, never
used numbers, nor felt the need for them.We may ask, why then did the
Sumerians on the other side of the world feel the need for simple
mathematics? The answer of course, was because they lived in cities which
required organizing. For example, grain needed to be stored and
determining how much each citizen received required arithmetic.
Egyptians loved all big things, such as big buildings, big statues and big
armies. They developed numbers of drudgery for everyday labor and large
numbers for aristocrats, such as a thousand, ten thousand and even a

15. million.The Egyptians transformation of using “one” from counting things to
measuring things was of great significance.
Their enthusiasm for building required accurate measurements so they
defined their own version of “one.” A cubit was defined as the length of a
mans arm from elbow to finger tips plus the width of his palm. Using this
standardized measure of “one” the Egyptians completed vast construction
projects, such as their great pyramids, with astonishing accuracy.
Two and a half thousand years ago, in 520 BCE, Pythagorus founded his
vegetarian school of math in Greece. Pythagorus was intrigued by whole
numbers,noticing that pleasing harmonies are combinations of whole
numbers. Convinced that the number one was the basis of the universe, he
tried to make all three sides of a triangle an exact number of units, a feat
which he was not able to accomplish. He was thus defeated by his own
favorite geometrical shape, one for which he would be forever famous.
His Pythagorean theorem has been credited to him, even though ancient
Indian texts, the Sulva Sutras (800 BCE) and the Shatapatha Brahmana (8th
to 6th centuries BCE) prove that this theorem was known in India some two
thousand years before his birth.
Later in the third century BCE, Archimedes, the renowned Greek scientist,
who loved to play games with numbers, entered the realm of the
unimaginable, trying to calculate such things as how many grains of sand
would fill the entire universe. Some of these intellectual exercises proved to
be useful, such as turning a sphere into a cylinder. His formula was later
used to take a globe and turn it into a flat map.
Romans invading Greece were interested in power, not abstract
mathematics. They killed Archimedes in 212 BCE and thereby impeded the
development of mathematics. Their system of Roman numerals was too
complicated for calculating, so actual counting had to be done on a counting
board, an early form of the abacus.
Although the usage of the Roman numeral system spread all over Europe
and remained the dominant numeral system for more than five hundred
years, not a single Roman mathematician is celebrated today. The Romans
were more interested in using numbers to record their conquests and count
dead bodies.

Numbers in Early India

16. In India, emphasis was not on military organization but in finding
enlightenment. Indians, as early as 500 BCE, devised a system of different
symbols for every number from one to nine, a system that came to be called
Arabic numerals, because they spread first to Islamic countries before
reaching Europe centuries later.
What is historically known goes back to the days of the Harappan civilization
(2,600-3,000 BCE). Since this Indian civilization delved into commerce and
cultural activities, it was only natural that they devise systems of weights
and measurements. For example a bronze rod marked in units of 0.367
inches was discovered and points to the degree of accuracy they demanded.
Evidently,such accuracy was required for town planning and construction
projects.Weights corresponding to units of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10,
20, 50, 100, 200 and 500 have been discovered and they obviously played
important parts in the development of trade and commerce.
It seems clear from the early Sanskrit works on mathematics that the
insistent demand of the times was there, for these books are full of problems
of trade and social relationships involving complicated calculations. There
are problems dealing with taxation, debt and interest, problems of
partnership, barter and exchange, and the calculation of the fineness of
gold. The complexities of society, government operations and extensive
trade required simpler methods of calculation.
Earliest Indian Literary and Archaeological References
When we discuss the numerals of today’s decimal number system we usuall y
refer to them as “Arabian numbers.” Their origin, however, is in India, where
they were first published in the Lokavibhaga on the 28th of August 458
AD.This Jain astronomical work, Lokavibhaga or “Parts of the Universe,” is
the earliest document clearly exhibiting familiarity with the decimal system.
One section of this same work gives detailed astronomical observations that
confirm to modern scholars that this was written on the date it claimed to be
written: 25 August 458 CE (Julian calendar). As Ifrah
2
points out, this
information not only allows us to date the document with precision, but also
proves its authenticity. Should anyone doubt this astronomical information,
it should be pointed out that to falsify such data requires a much greater
understanding and skill than it does to make the original calculations.
The origin of the modern decimal-based place value system is ascribed to
the Indian mathematician Aryabhata I, 498 CE. Using Sanskrit numeral
words for the digits, Aryabhata stated “Sthanam sthanam dasa gunam” or
“place to place is ten times in value.”The oldest record of this value place

17. assignment is in a document recorded in 594 CE, a donation charter of
Dadda III of Sankheda in the Bharukachcha region.
The earliest recorded inscription of decimal digits to include the symbol for
the digit zero, a small circle, was found at the Chaturbhuja Temple at
Gwalior, India, dated 876 CE.This Sanskrit inscription states that a garden
was planted to produce flowers for temple worship and calculations were
needed to assure they had enough flowers. Fifty garlands are mentioned
(line 20), here 50 and 270 are written with zero. It is accepted as the
undisputed proof of the first use of zero.
The usage of zero along with the other nine digits opened up a whole new
world of science for the Indians. Indeed Indian astronomers were centuries
ahead of the Christian world.The Indian scientists discovered that the earth
spins on its axis and moves around the sun, a fact that Copernicus in Europe
didn’t understand until a thousand years later—a discovery that he would
have been persecuted for, had he lived longer.
From these and other sources there can be no doubt that our modern
system of arithmetic—differing only in variations on the symbols used for the
digits and minor details of computational schemes—originated in India at
least by 510 CE and quite possibly by 458 CE.
The first sign that the Indian numerals were moving west comes from a
source which predates the rise of the Arab nations. In 662 AD Severus
Sebokht, a Nestorian bishop who lived in Keneshra on the Euphrates river,
wrote regarding the Indian system of calculation with decimal numerals:
“ ... more ingenious than those of the Greeks and the Babylonians, and of
their valuable methods of calculation which surpass description...”
3

This passage clearly indicates that knowledge of the Indian number system
was known in lands soon to become part of the Arab world as early as the
seventh century. The passage itself, of course, would certainly suggest that
few people in that part of the world knew anything of the system. Severus
Sebokht as a Christian bishop would have been interested in calculating the
date of Easter (a problem to Christian churches for many hundreds of
years). This may have encouraged him to find out about the astronomy
works of the Indians and in these, of course, he would find the arithmetic of
the nine symbols.
The Decimal Number System

18. The Indian numerals are elements of Sanskrit and existed in several variants
well before their formal publication during the late Gupta Period (c. 320-540
CE). In contrast to all earlier number systems, the Indian numerals did not
relate to fingers, pebbles, sticks or other physical objects.

The development of this system hinged on three key abstract (and certainly
non-intuitive) principles: (a) The idea of attaching to each basic figure
graphical signs which were removed from all intuitive associations, and did
not visually evoke the units they represented; (b) The idea of adopting the
principle according to which the basic figures have a value which depends on
the position they occupy in the representation of a number; and (c) The idea
of a fully operational zero, filling the empty spaces of missing units and at
the same time having the meaning of a null number.
4

The great intellectual achievement of the Indian number system can be
appreciated when it is recognized what it means to abandon the
representation of numbers through physical objects. It indicates that Indian
priest-scientists thought of numbers as an intellectual concept, something
abstract rather than concrete. This is a prerequisite for progress in
mathematics and science in general, because the introduction of irrational
numbers such as “pi,” the number needed to calculate the area inside a
circle, or the use of imaginary numbers is impossible unless the link between
numbers and physical objects is broken.
The Indian number system is exclusively a base 10 system, in contrast to
the Babylonian (modern-day Iraq) system, which was base 60; for example,
the calculation of time in seconds, minutes and hours. By the middle of the
2nd millennium BC, the Babylonian mathematics had a sophisticated
sexagesimal positional numeral system (based on 60, not 10). Despite the

19. invention of zero as a placeholder, the Babylonians never quite discovered
zero as a number.
The lack of a positional value (or zero) was indicated by a space between
sexagesimal numerals.They added the “space” symbol for the zero in about
400 BC. However, this effort to save the first place-value number system did
not overcome its other problems and the rise of Alexandria spelled the end
of the Babylonian number system and its cuneiform (hieroglyphic like)
numbers.
It is remarkable that the rise of a civilization as advanced as Alexandria also
meant the end of a place-value number system in Europe for nearly 2,000
years. Neither Egypt nor Greece nor Rome had a place-value number
system, and throughout medieval times Europe used the absolute value
number system of Rome (Roman Numerals). This held b ack the
development of mathematics in Europe and meant that before the period of
Enlightenment of the 17th century, the great mathematical discoveries were
made elsewhere in East Asia and in Central America.
The Mayans in Central America independently invented zero in the fourth
century CE.Their priest-astronomers used a snail-shell-like symbol to fill
gaps in the (almost) base-20 positional ‘long-count’ system they used to
calculate their calendar. They were highly skilled mathematicians,
astronomers, artists and architects. However, they failed to make other key
discoveries and inventions that might have helped their culture survive. The
Mayan culture collapsed mysteriously around 900 CE. Both the Babylonians
and the Mayans found zero the symbol, yet missed zero the number.
Although China independently invented place value, they didn’t make the
leap to zero until it was introduced to them by a Buddhist astronomer from
India in 718 CE.
Zero becomes a real numbe r
The concept of zero as a number and not merely a symbol for separation is
attributed to India where by the 9th century CE practical calculations were
carried out using zero, which was treated like any other number, even in the
case of division.
The story of zero is actually a story of two zeroes: zero as a symbol to
represent nothing and zero as a number that can be used in calculations and
has its own mathematical properties.
It has been commented that in India, the concept of nothing is important in
its early religion and philosophy and so it was much more natural to have a

20. symbol for it than for the Latin (Roman) and Greek systems. The rules for
the use of zero were written down first by Brahmagupta, in his book
“Brahmasphutha Siddhanta” (The Opening of the Universe) in the year 628
CE. Here Brahmagupta considers not only zero, but negative numbers, and
the algebraic rules for the elementary operations of arithmetic with such
numbers.
“The importance of the creation of the zero mark can never be
exaggerated.This giving to airy nothing, not merely a local habitation and a
name, a picture, a symbol, but helpful power, is the characteristic of the
Hindu race from whence it sprang. It is like coining the Nirvana into
dynamos. No single mathematical creation has been more potent for the
general on-go of intelligence and power.” - G. B. Halsted
5

A very important distinction for the Indian symbol for zero, is that, unlike
the Babylonian and Mayan zero, the Indian zero symbol came to be
understood as meaning nothing.
As the Indian decimal zero and its new mathematics spread from the Arab
world to Europe in the Middle Ages, words derived from sifr and zephyrus
came to refer to calculation, as well as to privileged knowledge and secret
codes. Records show that the ancient Greeks seemed unsure about the
status of zero as a number.They asked themselves,“How can nothing be
something?” This lead to philosophical and, by the Medieval period, religious
arguments about the nature and existence of zero and the vacuum.
The word “zero” came via the French word zéro, and cipher came from the
Arabic word safira which means “it was empty.” Also sifr, meaning “zero” or
“nothing,” was the translation for the Sanskrit word sunya, which means
void or empty.
The number zero was especially regarded with suspicion in Europe, so much
so that the word cipher for zero became a word for secret code in modern
usage. It is very likely a linguistic memory of the time when using decimal
arithmetic was deemed evidence of dabbling in the occult, which was
potentially punishable by the all-powerful Catholic Church with death.
6

WHAT IS THE ORIGIN OF
NUMBERS?

21. The numbers that are used worldwide nowadays are basically Arabic numbers. They
were actually developed by Arabic Muslim scientists who revised the Indian version of
numbers that contains only nine numbers. That took place during the 8th century (771
A.D) when an Indian Gastronomist came to the Almansour royal palace with a book –
famous at that time – about astronomy and mathematics called “Sod hanta” written by
Brahma Jobta around 626 A.D. Almansour ordered to translate the book into Arabic and
explore more sciences.
There were different forms in the Indian version of numbers, Arabs kept some of these
forms and changed others to create their own vision of numbers which was used in the
Middle East and mainly in Baghdad. Thanks to Al Khawarizmi (Algoritmi), Arabic
numbers took their final form. At the beginning, they were not widely spread but they
became known in the Maghreb and Andalusia. Europe then adopted these numbers
because of their practicality in comparison with Roman numbers. Now they are used
worldwide.
1 -Indian numbers:



2-Arabic numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9
In designing the Arab numbers, Al Khawarizmi based his choice of a particular form on
the number of angles that each number should contain. For instance, the number one
contains only one angle, number two has two angles, and number three includes three
angles, ects…
This picture clarifies the original forms of the Arab numbers, in each angle contains a
dot:

22. These numbers were later modified until they reached the present forms in which we
use them now. But the genius invention that Muslim scientists brought to us is the zero
(as it contains no angles).
The first usage of the zero dates back to 873 A.D, but the first Indian zero was
registered around 876 A.D.
The numbers used worldwide nowadays are all Arabic numbers not only because of
their beautiful forms, but also because of their practicality . Indeed, unlike Indian
numbers, Arabic numbers make a clear distinction between the zero and the dot so that
no confusion would be made while reading numbers.

Classification of Numbers
This classification of Numbers represents the most accepted elementary classification,
and is useful in computing sense.

Class Symbol Description

23. Natural Number
Natural numbers are defined as non-negative counting
numbers: = { 0, 1, 2, 3, 4, ... }. Some exclude 0
(zero) from the set: * = \{0} = { 1, 2, 3, 4, ...
}.
Integer
Integers extend by including the negative of
counting numbers:
= { ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ... }.
The symbol stands for Zahlen, the German word for
"numbers".
Rational Number

A rational number is the ratio or quotient of an integer
and another non-zero integer:
= {n/m | n, m ∈ , m ≠ 0 }.
E.g.: -100, -20¼, -1.5, 0, 1, 1.5, 1½ 2¾, 1.75, &c
Irrational Number Irrational numbers are numbers which cannot be
represented as fractions.
E.g.: √2, √3;, π, e.
Real Number
Real numbers are all numbers on a number line. The set
of is the union of all rational numbers and all
irrational numbers.
Imaginary Number An imaginary number is a number which square is a
negative real number, and is denoted by the symbol i,
so that i
2
= -1.
E.g.: -5i, 3i, 7.5i, &c.
In some technical applications, j is used as the symbol
for imaginary number instead of i.
Complex Number
A complex number consists of two part, real number
and imaginary number, and is also expressed in the
form a + bi (i is notation for imaginary part of the
number).
E.g.: 7 + 2i





Definition of Real Numbers

24. Properties, Examples, and Non Examples
A Real Definition :
The definition in math text books of real numbers is often not helpful to the average
person who is trying to gain an introductory and intuitive sense of what a real
number.
Real numbers are just the numbers on the number line.

It is the easiest way to think of them. Basically, if you can put the number in
question on an infinitely big number line, then it is a real number. Also, you have to
be add ,subtract ,multiply, divide that number in a way that is consistent with the
number line. They include many types of numbers:
Types of Real Numbers with examples
 Rational Numbers -- in other words all integers , fractions and decimals
(including repeating decimals)
o ex: 2,3 -2, ½, -¾ , .34
 Irrational Numbers
o , , yes, irrational numbers can be ordered and put on a
number line, we know that comes before
Properties of Real Numbers
 Real numbers can be ordered (this is not true, for instance, of imaginary
numbers )
 They can be added, subtracted , multiplied and divided by nonzero numbers
in an ordered way. So what does that mean? Basically it means that
comes before on the number line and that they both come
before . We know that this fact is true for rational and irrational
numbers. Think about the rational numbers 3 and 5, we know that we can
order 3 and 5 as follows. 3 comes before 5 and both numbers come before
8(3+5) .




Even and Odd Numbers

25. Even numbers can be divided evenly into groups of two. The number four can be
divided into two groups of two.
Odd numbers can NOT be divided evenly into groups of two. The number five can
be divided into two groups of two and one group of one.
Even numbers always end with a digit of 0, 2, 4, 6 or 8.
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 are even numbers.
Odd numbers always end with a digit of 1, 3, 5, 7, or 9.
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31 are odd numbers.

1
Odd
2
Even
3
Odd
4
Even
5
Odd
6
Even
7
Odd
8
Even
9
Odd
10
Even
11
Odd
12
Even
1
Odd
3
Odd
5
Odd
7
Odd
9
Odd
11
Odd
2
Even
4
Even
6
Even
8
Even
10
Even
12
Even

Odd Number
An odd number is an integer of the form , where is an integer. The odd numbers are therefore ...,
, , 1, 3, 5, 7, ... (OEIS A005408), which are also the gnomonic numbers. Integers which are not odd are
called even.
Odd numbers leave a remainder of 1 when divided by two, i.e., the congruence holds for odd . The
oddness of a number is called itsparity, so an odd number has parity 1, while an even number has parity 0.
The generating function for the odd numbers is

The product of an even number and an odd number is always even, as can be seen by writing

which is divisible by 2 and hence is even.

Prime and Composite Numbers

26. A prime number is a whole number that only has two factors which are itself and
one. A composite number has factors in addition to one and itself.
The numbers 0 and 1 are neither prime nor composite.
All even numbers are divisible by two and so all even nu mbers greater than two
are composite numbers.
All numbers that end in five are divisible by five. Therefore all numbers that end
with five and are greater than five are composite numbers.
The prime numbers between 2 and 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

http://gradestack.com/Complete-CAT-Prep/Number-System/Prime-Numbers-and/19133-3880-35700-
study-wtw

27. Chapter 3:
Indices,
Surds &
Logarithm

28. Exponents

The exponent of a number says how many times to use the number in a
multiplication.
In 8
2
the "2" says to use 8 twice in a multiplication,
so 8
2
= 8 × 8 = 64
In words: 8
2
could be called "8 to the power 2" or "8 to the second power", or
simply "8 squared"
Exponents are also called Powers or Indices.
Some more examples:
Example: 5
3
= 5 × 5 × 5 = 125
 In words: 5
3
could be called "5 to the third power", "5 to the power 3" or
simply "5 cubed"
Example: 2
4
= 2 × 2 × 2 × 2 = 16
 In words: 2
4
could be called "2 to the fourth power" or "2 to the power 4" or
simply "2 to the 4th"
Exponents make it easier to write and use many multiplications
Example: 9
6
is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9

29. You can multiply any number by itself as many times as you want
using exponents.
Try here: 3 4

3
4
= 3 × 3 × 3 × 3 = 81
© 2015 MathsIsFun.com v0.81
In General
So in general:
a
n
tells you to multiply a by itself,
so there are n of those a's:


Other Way of Writing It
Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is
easy to type.
Example: 2^4 is the same as 2
4

 2^4 = 2 × 2 × 2 × 2 = 16
Negative Exponents
Negative? What could be the opposite of multiplying?
Dividing!

30. A negative exponent means how many times to divide one by the number.
Example: 8
-1
= 1 ÷ 8 = 0.125
You can have many divides:
Example: 5
-3
= 1 ÷ 5 ÷ 5 ÷ 5 = 0.008
But that can be done an easier way:
5
-3
could also be calculated like:
1 ÷ (5 × 5 × 5) = 1/5
3
= 1/125 = 0.008
In General

That last example showed an easier way to handle negative
exponents:
 Calculate the positive exponent (a
n
)
 Then take the Reciprocal (i.e. 1/a
n
)
More Examples:
Negative Exponent Reciprocal of Positive Exponent Answer
4
-2
= 1 / 4
2
= 1/16 = 0.0625
10
-3
= 1 / 10
3
= 1/1,000 = 0.001
(-2)
-3
= 1 / (-2)
3
= 1/(-8) = -0.125

31. What if the Exponent is 1, or 0?
1

If the exponent is 1, then you just have the number itself
(example 9
1
= 9)

0
If the exponent is 0, then you get 1 (example 9
0
= 1)


But what about 0
0
? It could be either 1 or 0, and so people say it
is "indeterminate".
It All Makes Sense
My favorite method is to start with "1" and then multiply or divide as many
times as the exponent says, then you will get the right answer, for example:
Example: Powers of 5
.. etc..

5
2
1 × 5 × 5 25
5
1
1 × 5 5
5
0
1 1
5
-1
1 ÷ 5 0.2
5
-2
1 ÷ 5 ÷ 5 0.04
.. etc..

32. If you look at that table, you will see that positive, zero or negative exponents
are really part of the same (fairly simple) pattern.
Be Careful About Grouping
To avoid confusion, use parentheses () in cases like this:
With () : (-2)
2
= (-2) × (-2) = 4
Without () : -2
2
= -(2
2
) = - (2 × 2) = -4

With () : (ab)
2
= ab × ab
Without () : ab
2
= a × (b)
2
= a × b × b

Indices & the Law of Indices
Introduction
Indices are a useful way of more simply expressing large numbers. They also present us with many
useful properties for manipulating them using what are called the Law of Indices.
What are Indices?
The expression 2
5
is defined as follows:

We call "2" the base and "5" the index.
Law of Indices
To manipulate expressions, we can consider using the Law of Indices. These laws only apply to
expressions with the same base, for example, 3
4
and 3
2
can be manipulated using the Law of
Indices, but we cannot use the Law of Indices to manipulate the expressions 3
5
and 5
7
as their base
differs (their bases are 3 and 5, respectively).
Six rules of the Law of Indices
Rule 1:

33. Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.
An Example:
Simplify 2
0
:

Rule 2:

An Example:
Simplify 2
-2
:

Rule 3:
To multiply expressions with the same base, copy the base and add the indices.
An Example:
Simplify : (note: 5 = 5
1
)

Rule 4:
To divide expressions with the same base, copy the base and subtract the indices.
An Example:
Simplify :

Rule 5:
To raise an expression to the nth index, copy the base and multiply the indices.
An Example:
Simplify (y
2
)
6
:

Rule 6:

An Example:
Simplify 125
2/3
:

34. You have now learnt the important rules of the Law of Indices and are ready to try out some
examples!

Unit 3 Section 2 : Laws of Indices
There are three rules that should be used when working with indices:
When m and n are positive integers,
1. a
m
× a
n
= a
m + n

2. a
m
÷ a
n
= a
m – n
or
a
m

a
n


= a
m – n
(m ≥ n)
3. (a
m
)
n
= a
m × n


These three results are logical consequences of the definition of a
n
, but really need a
formal proof. You can 'verify' them with particular examples as below, but this is not
a proof:
2
7
× 2
3
= (2 × 2 × 2 × 2 × 2 × 2 × 2) × (2 × 2 × 2)


= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2


= 2
10
(here m = 7, n = 3 and m + n = 10)

35. or,
2
7
÷ 2
3
=
2 × 2 × 2 × 2 × 2 × 2 × 2
2 × 2 × 2



= 2 × 2 × 2 × 2


= 2
4
(again m = 7, n = 3 and m – n = 4)
Also,
(2
7
)
3
=
2
7
× 2
7
×
2
7


= 2
21
(using rule 1) (again m = 7, n = 3 and m × n = 21)
The proof of the first rule is given below:
Proof
a
m
× a
n
= a × a × ... × a

m of these
× a × a × ... × a

n of these
= a × a × ... × a × a × a × ... × a

(m+n) of these
= a
m+n

The second and third rules can be shown to be true for all positive
integers m and n in a similar way.
We can see an important result using rule 2:

36. x
n

x
n


= x
n – n
=
x
0

but
x
n

x
n


= 1, so
x
0
= 1
This is true for any non-zero value of x, so, for example, 3
0
= 1, 27
0
= 1 and 1001
0
=
1.
Example 1
Fill in the missing numbers in each of the following expressions:
(a) 2
4
× 2
6
= 2

(b) 3
7
× 3
9
= 3

(c) 3
6
÷ 3
2
= 3

(d) (10
4
)
3
= 10

Example 2
Simplify each of the following expressions so that it is in the form a
n
, where n is a
number:
(a) a
6
× a
7


(b)
a
4
× a
2

a
3



(c) (a
4
)
3

37. Exercises
Work out the answers to the questions below and fill in the boxes. Click on
the button to find out whether you have answered correctly. If you are right
then will appear and you should move on to the next question.
If appears then your answer is wrong. Click on to clear your
original answer and have another go. If you can't work out the right answer then
click on to see the answer.
Question 1
Fill in the missing numbers:
(a)
2
3
× 2
7
= 2

(b)
3
6
× 3
5
= 3

(c)
3
7
÷ 3
4
= 3

(d)
8
3
× 8
4
= 8

(e)
(3
2
)
5
= 3

(f)
(2
3
)
6
= 2

(g)
3
6

3
2


= 3

(h)
4
7

4
2


= 4

Question 2
Fill in the missing numbers:

38. (a)
a
3
× a
2
= a

(b)
b
7
÷ b
2
= b

(c)
(b
2
)
5
= b

(d)
b
6
× b
4
= b

(e)
(z
3
)
9
= z

(f)
q
16

q
7


= q

Question 3
Explain why 9
4
= 3
8
.
( ) =
×
=
Question 4
Calculate:
(a) 3
0
+ 4
0



(b) 6
0
× 7
0



(c) 8
0
– 3
0



(d) 6
0
+ 2
0
– 4
0



Question 5
Fill in the missing numbers:

39. (a)
3
6
× 3 = 3
17


(b)
4
6
× 4 = 4
11


(c)
a
6

a

= a
4


(d)
(z )
6
= z
18


(e)
(a
19
) = a
95


(f)
p
16
÷ p = p
7


(g)
(p )
8
= p
40


(h)
q
13
÷ q = q

Question 6
Calculate:
(a)
2
3

2
2


+ 3
0


(b)
3
4

3
3


– 3
0


(c)
5
4

5
2


+
6
2

6

40. (d)
7
7

7
5


–
5
9

5
7



(e)
10
8

10
5


–
5
6

5
3



(f)
4
17

4
14


–
4
13

4
11



Question 7
Fill in the missing numbers in each of the following expressions:
(a)
8
2
= 2

(b)
81
3
= 9 = 3

(c)
25
6
= 5

(d)
4
7
= 2

(e)
125
4
= 5

(f)
1000
6
= 10

(g)
81 =
4


(h)
256 =
4
=
8


Question 8
Fill in the missing numbers in each of the following expressions:

41. (a) 8 × 4 =
2 × 2
=
2

(b) 25 × 625 =
5 × 5
=
5

(c)
243
9

=
3
3

=
3

(d)
128
16

=
2
2

=
2

Question 9
Is each of the following statements true or false?
(a) 3
2
× 2
2
= 6
4


(b) 5
4
× 2
3
= 10
7


(c)
6
8

2
8


= 3
8


(d)
10
8

5
6


= 2
2


Question 10
Complete each expression:
(a)
(2
6
× 2
3
)
4
= (2 )
4
= 2

42. (b)

3
6

3
2



5

= (3 )
5
= 3


(c)

2
3
× 2
4

2
7



4

= (2 )
4
= 2


(d)

3
2
× 9
3
3



4

= (3 )
4
= 3


(e)

6
2
× 6
8

6
3



4

= (6 )
4
= 6


(f)

7
8

7
2
× 7
3



5

= (7 )
5
= 7


Uses of Index Numbers
The main uses of index numbers are given below:
o Index numbers are used in the fields of commerce, meteorology, labour, industrial, etc.
o The index numbers measure fluctuations during intervals of time,
group differences ofgeographical position of degree etc.
o They are used to compare the total variations in the prices of different commodities in
which the unit of measurements differs with time and price etc.
o They measure the purchasing power of money.
o They are helpful in forecasting the future economic trends.
o They are used in studying difference between the comparable categories of
animals,persons or items.
o Index numbers of industrial production are used to measure the changes in the level of
industrial production in the country.

43. o Index numbers of import prices and export prices are used to measure the changes in
the trade of a country.
o The index numbers are used to measure seasonal variations and cyclical variations in a
time series.

Surds
When we can't simplify a number to remove a square root (or cube root etc)
then it is a surd.
Example: √2 (square root of 2) can't be simplified further so it is a surd
Example: √4 (square root of 4) can be simplified (to 2), so it is not a surd!
Have a look at some more examples:
Number Simplified As a Decimal
Surd or
not?
√2 √2 1.4142135...(etc) Surd
√3 √3 1.7320508...(etc) Surd
√4 2 2 Not a surd
√¼ ½ 0.5 Not a surd
3
√11
3
√11 2.2239800...(etc) Surd
3
√27 3 3 Not a surd
5
√3
5
√3 1.2457309...(etc) Surd

The surds have a decimal which goes on forever without repeating, and
are Irrational Numbers.

44. In fact "Surd" used to be another name for "Irrational",
but it is now used for a root that is irrational.

How did we get the word "Surd" ?
Well around 820 AD al-Khwarizmi (the Persian guy who we get the name
"Algorithm" from) called irrational numbers "'inaudible" ... this was later
translated to the Latin surdus ("deaf" or "mute")
Conclusion
 When it is a root and irrational, it is a surd.
 But not all roots are surds.


Surds
Introduction
Surds are numbers left in root form (√) to express its exact value. It has an infinite number of non-
recurring decimals. Therefore, surds are irrational numbers.
There are certain rules that we follow to simplify an expression involving surds. Rationalising the
denominatoris one way to simplify these expressions. It is done by eliminating the surd in the
denominator. This is shown in Rules 3, 5 and 6.
It can often be necessary to find the largest perfect square factor in order to simplify surds. The
largest perfect square factor is found by looking at any possible factors of the number that is being
square rooted. Lets say that you are looking at the square root of 242. Can you simplify this? Well, 2
x 121 is 242 and we can take the square root of 121 without leaving a surd (because we get 11).
Since we cannot take the square root of a larger number that can be multiplied by another to give
242 then we say that 121 is the largest perfect square factor.
Six Rules of Surds
Rule 1:
An Example:
Simplify :

45. Since , as 9 is the largest perfect square factor of 18.

Rule 2:

An Example:
Simplify

:

Rule 3:

By multiplying both the numberator and denominator by the denominator you can rationalise the
denominator.
An Example:
rationalise

:

Rule 4:
An Example:
Simplify :

46. Rule 5:

Following this rule enables you to rationalise the denominator.
An Example:
Rationalise

:

Rule 6:

Following this rule enables you to rationalise the denominator.
An Example:
Rationalise

:


Base b Logarithm
The base
b
logarithm of
x,

logbx,
is the power to which you need to raise
b
in order to get
x.

47. Symbolically,
logbx = y means by = x.
Logarithmic form

Exponential form
Notes
1.
logbx
is only defined if
b
and
x
are both positive, and
b ≠ 1.

2.
log10x
is called the common logarithm of
x,
and is sometimes written as
log x.

3.
logex
is called the natural logarithm of
x
and is sometimes written as
ln x.
Examples
The following table lists some exponential equations and their equivalent logarithmic form.
Exponentia
l Form
10
3
= 1,000 4
2
= 16 3
3
= 27 5
1
= 5 7
0
= 1 4
−2
= 1/16 25
1/2
= 5
Logarithmi
c Form
log10
1,000 =
3
log416 =
2
log327 =
3
log55 =
1
log71 =
0
log4(1/16) = −
2
log25
5 = 1/
2
Here are some for you to try
Exponential Form 10
2
= 100 3
−2
= 1/9
Logarithmic Form
log


=


log


=

48. Exponential Form
^ =

^ =

Logarithmic Form log31 = 0 log5(1/125) = − 3


Example 1 Calculating Logarithms by Hand
(a) log28 =
Power to which you need to raise
2
in order to get
8

=
3 Since
2
3
= 8
(b) log41 =
Power to which you need to raise
4
in order to get
1

=
0
Since
4
0
= 1
(c) log10
10,000 =
Power to which you need to raise
10
in order to get
10,000

=
4
Since
10
4
= 10,000
(d) log10
1/100 =
Power to which you need to raise
10
in order to get
1/100

=
−2
Since
10
−2
= 1/100
(e) log327 =

(f) log93 =

(g) log3(1/81) =


Algebraic Properties of Logarithms

49. The following identities hold for any positive
a ≠ 1
and any positive numbers
x
and
y.
Identity Example
(a)
loga(xy) = logax + logay
log216 = log28 + log22
(b)
loga
(


x
y


)
= logax − logay
log2
(


5
3


)
= log25 − log23
(c)
loga(x
r
) = rlogax
log2(6
5
) = 5log26
(d)
logaa = 1

loga1 = 0

log22 = 1
log31 = 0

(e)
loga
(


1
x


)
= − logax
log2
(


1
3


)
= − log23
(f)
logax =
log25 =

50. log x
log a

=
ln x
ln a


log 5
log 2

≈ 2.3219


Example 2 Using the Properties of Logarithms
Let
a = log 2,

b = log 3,
and
c = log 5.
Write the following in terms of
a,

b,
and
c.

Note If any answer you give is not simplified -- for instance, if you say
a + a
instead of
2a
-- it will be marked wrong.

Answer
(a)
log 6

log 2 + log 3 = a + b
(b) log 15

(c)
log 30

log 2 + log 3 + log 5 = a + b+c
(d) log 12

(e)
log 1.5

log 3 − log 2 = b − a
(f) log(1/9)

(g) log 32 log 2
5
= 5log 2 = 5a

51. (h) log(1/81)


Q Where do the identities come from?
A Roughly speaking, they are restatements in logarithmic form of the laws of
exponents.
Q Why is
logaxy = logax + logay
?
A Let
s = logax,
and
t = logay.
In exponential form, these equations say that
a
s
= x
and
a
t
= y.
Multiplying these two equations together gives
a
s
a
t
= xy,

that is,
a
s + t
= xy.
Rewriting this in logarithmic form gives
loga(xy) = s + t = logax + logay
as claimed.
Here is an intuitive way of thinking about it: Since logs are exponents, this identity
expresses the familiar law that the exponent of a product is the sum of the exponents.
The second logarithmic identity is shown in almost the identical way, and we leave it
for you for practice.
Q Why is
loga(x
r
) = rlogax
?
A Let
t = logax.
Writing this in exponential form gives
a
t
= x.

52. Raising this equation to the
r
th

power gives
a
rt
= x
r
.
Rewriting in logarithmic form gives
loga(x
r
) = rt = rlogax,
as claimed.
Identity (d) we will leave for you to do as practice.
Q Why is
loga(1/x) = − logax
?
A This follows from identities (b) and (d) (think about it). <>
Q Why is
logax =
log x
log a

=
ln x
ln a

?
A Let
s = logax.
In exponential form, this says that
a
s
= x.
Take the logarithm with base b of both sides, getting
logba
s
= logbx,
then use identity (c):
slogba = log bx,
so
s =
logbx
logba

Since logarithms are exponents, we can use them to solve equations where the
unknown is in the exponent.
Example 3 Solving for the Exponent

53. Solve the following equations for
x.
(a)
4
−x2 = 1/64.

(b)
5(1.1
2x + 3
) = 200
Solution We can solve both of these equations by translating from exponential form
to logarithmic form.
(a) Write the given equation in logarithmic form:
4
−x2
= 1/64 Exponential Form

log4(1/64) = − x
2
Logarithmic Form
Thus, −x
2
= log4(1/64) = − 3

giving x = ±3
1/2
.

(b) Before converting to logarithmic form, first divide both sides of the equation by 5:
5(1.1
2x + 3
) = 200

1.1
2x + 3
= 40 Exponential Form

log1.140 = 2x + 3 Logarithmic Form
This gives 2x + 3 = ln 40/ln 1.1 ≈ 38.7039, Identity (e)
so that x ≈ 17.8520.


You can now either go on and try the exericses in the exercise set for this topic.

PROPERTIES OF LOGARITHMS

Property 1: because .

54. Example 1: In the equation , the base is 14 and the exponent is 0.
Remember that a logarithm is an exponent, and the corresponding logarithmic
equation is where the 0 is the exponent.
Example 2: In the equation , the base is and the exponent is 0.
Remember that a logarithm is an exponent, and the corresponding logarithmic
equation is .
Example 3: Use the exponential equation to write a logarithmic
equation. The base x is greater than 0 and the exponent is 0. The corresponding
logarithmic equation is .
Property 2: because .
Example 4: In the equation , the base is 3, the exponent is 1, and the
answer is 3. Remember that a logarithm is an exponent, and the
corresponding logarithmic equation is .
Example 5: In the equation , the base is 87, the exponent is 1, and
the answer is 87. Remember that a logarithm is an exponent, and the
corresponding logarithmic equation is .
Example 6: Use the exponential equation to write a logarithmic
equation. If the base p is greater than 0, then .
Property 3: because .
Example 7: Since you know that , you can write the logarithmic
equation with base 3 as .
Example 8: Since you know that , you can write the logarithmic
equation with base 13 as .

55. Example 9: Use the exponential equation to write a logarithmic
equation with base 4. You can convert the exponential equation
to the logarithmic equation . Since the 16 can be written
as
, the equation can be written .
The above rules are the same for all positive bases. The most common bases are the
base 10 and the base e. Logarithms with a base 10 are called common logarithms,
and logarithms with a base e arenatural logarithms. On your calculator, the base 10
logarithm is noted by log, and the base e logarithm is noted by ln.
There are an infinite number of bases and only a few buttons on your calculator. You
can convert a logarithm with a base that is not 10 or e to an equivalent logarithm with
base 10 or e. If you are interested in a discussion on how to change the bases of a
logarithm, click on Change of Base.
For a discussion of the relationship between the graphs of logarithmic functions and
exponential functions, click on graphs.
Properties of Logarithms
1. loga (uv) = loga u + loga v 1. ln (uv) = ln u + ln v
2. loga (u / v) = loga u - loga v 2. ln (u / v) = ln u - ln v
3. loga u
n
= n loga u 3. ln u
n
= n ln u
The properties on the left hold for any base a.
The properties on the right are restatements of the general properties for the natural
logarithm.
Many logarithmic expressions may be rewritten, either expanded or condensed, using
the three properties above. Expanding is breaking down a complicated expression into
simpler components. Condensing is the reverse of this process.
Example 2.
Expanding an expression.

56. rewrite using exponential notation
property 3
property 1
Example 3.
Expanding an expression.

property 2
property 1
property 3
Example 4.
Condensing an expression.

property 3
property 1
property 2
Common Mistakes
 Logarithms break products into sums by property 1, but the logarithm of a
sum cannot be rewritten. For instance, there is nothing we can do to the
expression ln( x
2
+ 1).
 log u - log v is equal to log (u / v) by property 2, it is not equal to log u / log v.
Exercise 3:
(a) Expand the expression . Answer
(b) Condense the expression 3 log x + 2 log y - (1/2) log z. Answer

57. 4.3 - Properties of Logarithms
Change of Base Formula
One dilemma is that your calculator only has logarithms for two bases on it.
Base 10 (log) and base e (ln). What is to happen if you want to know the
logarithm for some other base? Are you out of luck?
No. There is a change of base formula for converting between different bases.
To find the log base a, where a is presumably some number other than 10
or e, otherwise you would just use the calculator,
Take the log of the argument divided by the log of the base.
loga x = ( logb x ) / ( logb a )
There is no need that either base 10 or base e be used, but since those are
the two you have on your calculator, those are probably the two that you're
going to use the most. I prefer the natural log (ln is only 2 letters while log is 3,
plus there's the extra benefit that I know about from calculus). The base that
you use doesn't matter, only that you use the same base for both the
numerator and the denominator.
loga x = ( log x ) / ( log a ) = ( ln x ) / ( ln a )
Example: log3 7 = ( ln 7 ) / ( ln 3 )
Logarithms are Exponents
Remember that logarithms are exponents, so the properties of exponents are
the properties of logarithms.
Multiplication
What is the rule when you multiply two values with the same base together
(x
2
* x
3
)? The rule is that you keep the base and add the exponents. Well,
remember that logarithms are exponents, and when you multiply, you're going
to add the logarithms.
The log of a product is the sum of the logs.

58. loga xy = loga x + loga y
Division
The rule when you divide two values with the same base is to subtract the
exponents. Therefore, the rule for division is to subtract the logarithms.
The log of a quotient is the difference of the logs.
loga (x/y) = loga x - loga y
Raising to a Power
When you raise a quantity to a power, the rule is that you multiply the
exponents together. In this case, one of the exponents will be the log, and the
other exponent will be the power you're raising the quantity to.
The exponent on the argument is the coefficient of the log.
loga x
r
= r * loga x
Melodic Mathematics
Some of the statements above are very melodious. That is, they sound good.
It may help you to memorize the melodic mathematics, rather than the
formula.
 The log of a product is the sum of the logs
 The sum of the logs is the log of the products
 The log of a quotient is the difference of the logs
 The difference of the logs is the log of the quotient
 The exponent on the argument is the coefficient of the log
 The coefficient of the log is the exponent on the argument
Okay, so the last two aren't so melodic.
Common Mistakes
I almost hesitate to put this section in here. It seems when I try to point out a
mistake that people are going to make, that more people make it.

59.  The log of a sum is NOT the sum of the logs. The sum of the logs is the
log of the product. The log of a sum cannot be simplified.
loga (x + y) ≠ loga x + loga y
 The log of a difference is NOT the difference of the logs. The difference
of the logs is the log of the quotient. The log of a difference cannot be
simplified.
loga (x - y) ≠ loga x - loga y
 An exponent on the log is NOT the coefficient of the log. Only when the
argument is raised to a power can the exponent be turned into the
coefficient. When the entire logarithm is raised to a power, then it can
not be simplified.
(loga x)
r
≠ r * loga x
 The log of a quotient is not the quotient of the logs. The quotient of the
logs is from the change of base formula. The log of a quotient is the
difference of the logs.
loga (x / y) ≠ ( loga x ) / ( loga y )

Logarithm
Logarithm is reverse of exponentiation. Exponential and logarithms are inverse functions of each
other. The relation between exponential function and logarithm is given below:

Here, a is known as base and x is exponent. We pronounce logay as "log of y at base a".

For example: Exponential expression 23=8 will be equivalent to logarithmic expression log28=3.

Types of Logarithms
There are two types of logarithms:
 Common Logarithm: A logarithm with base 10 is known as common logarithm. For
example: log104. We see log without any base very often. It means that the base is 10. For
 example: log104 or log 4 are same.
 Natural Logarithm: A logarithm with base e is known as natural logarithm. For
example: loge4. e is a constant whose value is approximately 2.178. Natural logarithms are
also represented as ln x.

60. Change of Base Formula
We can convert a logarithmic expression of one base into that of another base by using following
formula:

Following are the rules for operations of logarithms:
 loga(mn)=logam+logan
 $log_{a}(mn)=log_{a}m-log_{a}n$
 logamn=nlogam
Exponents follow following rules:
 loga1=0 as a0=1
 logaa=1 as a1=a
 logaax=x as ax=ax

Logarithms can be divided into two types:
 Common logarithms:
Logarithms of the base 10 are called common logarithms.
o log1025
o log1010
o log1016

 Natural logarithms:
Logarithms of the base eare called natural logarithms.
o loge10
o loge400
o loger

Did you know that logea can be represented as ln a?
https://brilliant.org/discussions/thread/types-of-exponents-2/

Exponential Functions: Introduction (page 1 of 5)

61. Exponential functions look somewhat similar to functions you have seen before, in that
they involve exponents, but there is a big difference, in that the variable is now the
power, rather than the base. Previously, you have dealt with such functions as f(x) = x
2
,
where the variable x was the base and the number 2 was the power. In the case of
exponentials, however, you will be dealing with functions such as g(x) = 2
x
, where the
base is the fixed number, and the power is the variable.
Let's look more closely at the function g(x) = 2
x
. To evaluate this function, we operate as
usual, picking values of x, plugging them in, and simplifying for the answers. But to
evaluate 2
x
, we need to remember how exponents work. In particular, we need to
remember that negative exponents mean "put the base on the other side of the fraction
line".

So, while positive x-values give us values like these:

62. ...negative x-values give us values like these:
Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved




Putting together the "reasonable" (nicely
graphable) points, this is our T-chart:




...and this is our graph:

63. You should expect exponentials to look
like this. That is, they start small —very
small, so small that they're practically
indistinguishable from "y = 0", which is
the x-axis— and then, once they start
growing, they grow faster and faster, so
fast that they shoot right up through the
top of your graph.
You should also expect that your T-chart
will not have many useful plot points. For
instance, forx = 4 and x = 5, the y-values
were too big, and for just about all the
negative x-values, the y-values were too
small to see, so you would just draw the line right along the top of the x-axis.
Note also that my axis scales do not match. The scale on the x-axis is much wider than
the scale on the y-axis; the scale on the y-axis is compressed, compared with that of
the x-axis. You will probably find this technique useful when graphing exponentials,
because of the way that they grow so quickly. You will find a few T-chart points, and
then, with your knowledge of the general appearance of exponentials, you'll do your
graph, with the left-hand portion of the graph usually running right along the x-axis.


You may have heard of the term "exponential growth". This "starting slow, but then
growing faster and faster all the time" growth is what they are referring to. Specifically,
our function g(x) above doubled each time we incremented x. That is, when x was
increased by 1 over what it had been, y increased to twice what it had been. This is the
definition of exponential growth: that there is a consistent fixed period over which the
function will double (or triple, or quadruple, etc; the point is that the change is always a
fixed proportion). So if you hear somebody claiming that the world population is
doubling every thirty years, you know he is claiming exponential growth.
Exponential growth is "bigger" and "faster" than polynomial growth. This means that, no
matter what the degree is on a given polynomial, a given exponential function will
eventually be bigger than the polynomial. Even though the exponential function may
start out really, really small, it will eventually overtake the growth of the polynomial,
since it doubles all the time.

64. For instance, x
10
seems
much "bigger" than 10
x
, and
initially it is:




But eventually 10
x
(in blue
below) catches up and
overtakes x
10
(at the red
circle below, where x is ten
and y is ten billion), and it's
"bigger" than x
10
forever
after:




Exponential functions always have some positive number other than 1 as the base. If
you think about it, having a negative number (such as –2) as the base wouldn't be very
useful, since the even powers would give you positive answers (such as "(–2)
2
= 4") and
the odd powers would give you negative answers (such as "(–2)
3
= –8"), and what would
you even do with the powers that aren't whole numbers? Also, having 0 or 1 as the base
would be kind of dumb, since 0 and 1 to any power are just0 and 1, respectively; what
would be the point? This is why exponentials always have something positive and other
than 1 as the base.

Logarithmic Series

Infinite series of various simple functions of the logarithm include



(1)



(2)

65. (3)



(4)
where is the Euler-Mascheroni constant and is the Riemann zeta function. Note that the first two of these are
divergent in the classical sense, but converge when interpreted as zeta-regularized sums.
SEE ALSO:

Logarithmic Series

In mathematics, the logarithmic function is main division. Now we are going to explain about how to help
to the students about logarithmic series. Basic logarithmic function is defined as function
of , in logarithmic series there is no series about logarithm function like ln(x), but there
is simple series in
In mathematics, the logarithmic function is main division. Now we are going to explain about how to help
to the students about logarithmic series. Basic logarithmic function is defined as function
of , in logarithmic series there is no series about logarithm function like ln(x), but there
is simple series in
That is
Here are two things that are sign change to plus and minus alternating for the logarithmic series.
Now we are going see about help to the students based on logarithmic series.
If a > 0, by Exponential Theorem
+ ............to infinity
putting a = 1 + x

66. If , by Exponential Theorem
............... to infinity
putting a = 1 + x
.. .. ...........................................to infinity.
By Binomial Theorem for any index
................... to infinity
Equating these two series

Equating coifficients of y on both sides,
..................... to infinity
...................................to infinity
so
...................................to infinity
this series is called Logarithmic Series

67. Some Basic Logarithmic Series
Back to Top

Logarithmic series for help to the students online:
1.
2.
3.
4.
5.
6.
These are the basic important logarithmic series.
Examples on Logarithmic Series
Back to Top


Below are some examples based on logarithmic series
Problem 1: To solve the logarithmic function of
Solution: Given function is
We would rewrite the function depended upon logarithm series.
Like,
Now the function is

Here just taking the series,

68. Answer:
Problem 2: To solve the logarithmic function of
Solution: Given function is
Here we can apply logarithmic law, that is
Now the function is
Here just taking the series,




Answer:
Logarithmic Series Practice Problems
Back to Top

Prepare some problems about logarithmic series for help:
Problem 1: To solve the logarithmic function of using the logarithmic series.
Answer
Problem 2: To solve the logarithmic function of ln(1+3x)^-2 using the logarithmic series.
Answer

69. Problem 3: To solve the logarithmic function of ln(1+x)^-5 using the logarithmic series.
Answer

Exponential Series

In this page, we are going to discuss about exponential series concept. If an exponential
function e
x
can be expressed as an infinite exponent series, then it is an exponent
series function, which is shown below. The exponential function in math is defined as e
x
,
where e is an any integer. For example, let us assume e
x
as an exponential function. Let
us take the value of x as zero, x = 0. Then, the solution is of the form of e
0
= 1.

Exponential Series Expansion

Exponential series formula is of the form,
ex=1+x1!+x22!+x33!+......=∑∞n=0xnn!
From the given exponential series formula, we can port value for the exponent 'x' as an
infinite series like the given series of e
x
. If the exponent function of x is n, then the
coefficient function of x is 1n!.
Some of the exponential series formulas are as follows,
1. e−x=1−x1!+x22!−x33!+......
2. ex+e−x2=1+x22!+x44!+.........
3. ex−e−x2=x+x33!+x55!+.........
Exponential Fourier Series

If D is the amplitude, ω0 is the radian frequency [rad/s], ω02π is the frequency [hertz]
and T0=2πω0 is the period [sec.], then the exponential fourier series is defined as:

f(t)=∑∞n=−∞Dnejnω0t

Where, Dn=1T0∫T0f(t)ejnω0tdt

70. ω0=2πT0

and D0 = C0 = a0
Taylor Series Exponential

If we have a function y = f(x) = e
x
, then the taylor series of the function at any point x = a
for the given function is as follows:
f(x)=ex=ea+ea(x−a)+ea2!(x−a)2+ea3!(x−a)3+......

If a = 0, then e
0
= 1. Then, the exponential series is e
x
= 1+x+x22!+x33!+......

Some expansion of exponential series with the help of Taylor series is as follows:
1. e−x2=1−x21!+x42!−x63!+....
2. esinx=1+x+x22−x48−x515+.....
3. etanx=1+x+x22+x32+3x48+...... where IxI<π2
Exponential Theorem

If a > 0, then prove that ax=1+x(logea)+x22!(logea)2+......

Proof:
Exponential series is ex=1+x1!+x22!+x33!+...... ...................(i)

Now, let us assume that if logea=b, then eb=a
Put x = bx in equation (i), then we get

ebx=1+bx+(bx)22!+......

eb(x)=1+bx+(bx)22!+......

Replace e
bx
by a
x
, then

ax=1+x(logea)+x22!(logea)2+......

Exponential Series Examples
Given below are some of the examples on exponential series.

71. Solved Examples
Question 1: Expand the power series of given exponential function e
2x
.
Solution:
Given exponential function is e
2x

We know power series formula for e
x
:
ex=1+x+x22!+x33!+........
Now, applying the above formula, we get
e2x=1+(2x)+(2x)22!+(2x)33!+......
= 1+2x+4x22!+8x36+16x424+.......

= 1+2x+2x2+43x3+23x4+......
Thus, the power series of e
2x
is 1+2x+2x2+43x3+23x4+......

Question 2: Expand the power series of the exponential function e
- x
.
Solution:
The exponential function is e
- x

We know power series formula for e
x
:
ex=1+x+x22!+x33!+.........
Now, applying the above formula, we get
e−x=1+−x1!+(−x)22!+(−x)33!+........

= 1+−x1+(−x)22+(−x)36+........

= 1−x+x22−x36+.......
Thus, the power series of e
- x
is 1−x+x22−x36+.......

72. Chapter 4:
Set
Theory

73. Set theory


Written by: Robert R. Stoll
Set theory, branch of mathematics that deals with the properties of well-defined
collections of objects, which may or may not be of a mathematical nature, such as
numbers or functions. The theory is less valuable in direct application to ordinary
experience than as a basis for precise and adaptable terminology for
the definitionof complex and sophisticated mathematical concepts.

Between the years 1874 and 1897, the German mathematician and logician Georg
Cantor created a theory of abstract sets of entities and made it into a
mathematical discipline. This theory grew out of his investigations of some
concrete problems regarding certain types of infinite sets of real numbers. A set,
wrote Cantor, is a collection of definite, distinguishable objects of perception or
thought conceived as a whole. The objects are called elements or members of the
set.
The theory had the revolutionary aspect of treating infinite sets as mathematical
objects that are on an equal footing with those that can be constructed in a finite
number of steps. Since antiquity, a majority of mathematicians had carefully
avoided the introduction into their arguments of the actual infinite (i.e., of sets
containing aninfinity of objects conceived as existing simultaneously, at least in
thought). Since this attitude persisted until almost the end of the 19th century,
Cantor’s work was the subject of much criticism to the effect that it dealt with
fictions—indeed, that it encroached on the domain of philosophers and violated
the principles of religion. Once applications to analysis began to be found,
however, attitudes began to change, and by the 1890s Cantor’s ideas and results
were gaining acceptance. By 1900, set theory was recognized as a distinct branch
of mathematics.
At just that time, however, several contradictions in so-called naive set theory
were discovered. In order to eliminate such problems, an axiomatic basis was
developed for the theory of sets analogous to that developed for
elementary geometry. The degree of success that has been achieved in this

74. development, as well as the present stature of set theory, has been well
expressed in the Nicolas Bourbaki Éléments de mathématique (begun 1939;
“Elements of Mathematics”): “Nowadays it is known to be possible, logically
speaking, to derive practically the whole of known mathematics from a single
source, The Theory of Sets.”

Introduction to naive set theory
Fundamental set concepts
In naive set theory, a set is a collection of objects (called members or elements)
that is regarded as being a single object. To indicate that an object x is a member
of a setA one writes x ∊ A, while x ∉ A indicates that x is not a member of A. A set
may be defined by a membership rule (formula) or by listing its members within
braces. For example, the set given by the rule “prime numbers less than 10” can
also be given by {2, 3, 5, 7}. In principle, any finite set can be defined by an explicit
list of its members, but specifying infinite sets requires a rule or pattern to
indicate membership; for example, the ellipsis in {0, 1, 2, 3, 4, 5, 6, 7, …} indicates
that the list of natural numbers N goes on forever. The empty (or void, or null)
set, symbolized by {} or Ø, contains no elements at all. Nonetheless, it has the
status of being a set.
A set A is called a subset of a set B (symbolized by A ⊆ B) if all the members
of A are also members of B. For example, any set is a subset of itself, and Ø is a
subset of any set. If both A ⊆ B and B ⊆ A, then A and B have exactly the same
members. Part of the set concept is that in this case A = B; that is, A and B are the
same set.

OPERATIONS ON SETS

The symbol ∪ is employed to denote the union of two sets. Thus, the set A ∪ B—
read “A union B” or “the union of A and B”—is defined as the set that consists of
all elements belonging to either set A or set B (or both). For example, suppose
that Committee A, consisting of the 5 members Jones, Blanshard, Nelson, Smith,
and Hixon, meets with Committee B, consisting of the 5 members Blanshard,
Morton, Hixon, Young, and Peters. Clearly, the union of Committees A and B must

75. then consist of 8 members rather than 10—namely, Jones, Blanshard, Nelson,
Smith, Morton, Hixon, Young, and Peters.
The intersection operation is denoted by the symbol ∩. The set A ∩ B—read
“Aintersection B” or “the intersection of A and B”—is defined as the set
composed of all elements that belong to both A and B. Thus, the intersection of
the two committees in the foregoing example is the set consisting of Blanshard
and Hixon.
If E denotes the set of all positive even numbers and O denotes the set of all
positive odd numbers, then their union yields the entire set of positive integers,
and their intersection is the empty set. Any two sets whose intersection is the
empty set are said to be disjoint.
When the admissible elements are restricted to some fixed class of objects U, U is
called the universal set (or universe). Then for any subset A of U,
the complement ofA (symbolized by A′ or U − A) is defined as the set of all
elements in the universe Uthat are not in A. For example, if the universe consists
of the 26 letters of the alphabet, the complement of the set of vowels is the set of
consonants.
In analytic geometry, the points on a Cartesian grid are ordered pairs (x, y) of
numbers. In general, (x, y) ≠ (y, x); ordered pairs are defined so that (a, b) = (c, d)
if and only if both a = c and b = d. In contrast, the set {x, y} is identical to the set
{y, x} because they have exactly the same members.
The Cartesian product of two sets A and B, denoted by A × B, is defined as the set
consisting of all ordered pairs (a, b) for which a ∊ A and b ∊ B. For example,
ifA = {x, y} and B = {3, 6, 9}, then A × B = {(x, 3), (x, 6), (x, 9), (y, 3), (y, 6), (y, 9)}.

RELATIONS IN SET THEORY

In mathematics, a relation is an association between, or property of, various
objects. Relations can be represented by sets of ordered pairs (a, b)
where a bears a relation to b. Sets of ordered pairs are commonly used to
represent relations depicted on charts and graphs, on which, for example,
calendar years may be paired with automobile production figures, weeks with
stock market averages, and days with average temperatures.

76. A function f can be regarded as a relation between each object x in its domain and
the value f(x). A function f is a relation with a special property, however: each x is
related by f to one and only one y. That is, two ordered pairs (x, y) and (x, z)
in fimply that y = z.
A one-to-one correspondence between sets A and B is similarly a pairing of each
object in A with one and only one object in B, with the dual property that each
object in B has been thereby paired with one and only one object in A. For
example, ifA = {x, z, w} and B = {4, 3, 9}, a one-to-one correspondence can be
obtained by pairing x with 4, z with 3, and w with 9. This pairing can be
represented by the set {(x, 4), (z, 3), (w, 9)} of ordered pairs.
Many relations display identifiable properties. For example, in the relation “is the
same colour as,” each object bears the relation to itself as well as to some other
objects. Such relations are said to be reflexive. The ordering relation “less than or
equal to” (symbolized by ≤) is reflexive, but “less than” (symbolized by <) is not.
The relation “is parallel to” (symbolized by ∥) has the property that, if an object
bears the relation to a second object, then the second also bears that relation to
the first. Relations with this property are said to be symmetric. (Note that the
ordering relation is not symmetric.) These examples also have the property that
whenever one object bears the relation to a second, which further bears the
relation to a third, then the first bears that relation to the third—e.g.,
if a < b and b < c, then a < c. Such relations are said to be transitive.
Relations that have all three of these properties—reflexivity, symmetry,
andtransitivity—are called equivalence relations. In an equivalence relation, all
elements related to a particular element, say a, are also related to each other,
and they form what is called the equivalence class of a. For example, the
equivalence class of a line for the relation “is parallel to” consists of the set of all
lines parallel to it.

Essential features of Cantorian set theory

At best, the foregoing description presents only an intuitive concept of a set.
Essential features of the concept as Cantor understood it include: (1) that a set is
a grouping into a single entity of objects of any kind, and (2) that, given an

77. object x and a set A, exactly one of the statements x ∊ A and x ∉ A is true and the
other is false. The definite relation that may or may not exist between an object
and a set is called the membership relation.
A further intent of this description is conveyed by what is called the principle of
extension—a set is determined by its members rather than by any particular way
of describing the set. Thus, sets A and B are equal if and only if every element
in A is also in B and every element in B is in A; symbolically, x ∊ A implies x ∊ B and
vice versa. There exists, for example, exactly one set the members of which are 2,
3, 5, and 7. It does not matter whether its members are described as “prime
numbers less than 10” or listed in some order (which order is immaterial)
between small braces, possibly {5, 2, 7, 3}.
The positive integers {1, 2, 3, …} are typically used for counting the elements in a
finite set. For example, the set {a, b, c} can be put in one-to-one correspondence
with the elements of the set {1, 2, 3}. The number 3 is called the cardinal number,
or cardinality, of the set {1, 2, 3} as well as any set that can be put into a one-to-
one correspondence with it. (Because the empty set has no elements, its
cardinality is defined as 0.) In general, a set A is finite and its cardinality is n if
there exists a pairing of its elements with the set {1, 2, 3, … , n}. A set for which
there is no such correspondence is said to be infinite.
To define infinite sets, Cantor used predicate formulas. The phrase “x is a
professor” is an example of a formula; if the symbol x in this phrase is replaced by
the name of a person, there results a declarative sentence that is true or false.
The notation S(x) will be used to represent such a formula. The phrase “x is a
professor at university y and xis a male” is a formula with two variables. If the
occurrences of x and y are replaced by names of appropriate, specific objects, the
result is a declarative sentence that is true or false. Given any formula S(x) that
contains the letter x (and possibly others), Cantor’s principle of abstraction asserts
the existence of a set A such that, for each object x, x ∊ A if and only if S(x) holds.
(Mathematicians later formulated a restricted principle of abstraction, also known
as the principle of comprehension, in which self-referencing predicates, or S(A),
are excluded in order to prevent certain paradoxes.See below Cardinality and
transfinite numbers.) Because of the principle of extension, the
set A corresponding to S(x) must be unique, and it is symbolized by {x | S(x)},

78. which is read “The set of all objects x such that S(x).” For instance, {x | x is blue} is
the set of all blue objects. This illustrates the fact that the principle of abstraction
implies the existence of sets the elements of which are all objects having a certain
property. It is actually more comprehensive. For example, it asserts the existence
of a set B corresponding to “Either x is an astronaut or x is a natural number.”
Astronauts have no particular property in common with numbers (other than
both being members of B).

EQUIVALENT SETS

Cantorian set theory is founded on the principles of extension and abstraction,
described above. To describe some results based upon these principles, the
notion ofequivalence of sets will be defined. The idea is that two sets are
equivalent if it is possible to pair off members of the first set with members of the
second, with no leftover members on either side. To capture this idea in set-
theoretic terms, the set Ais defined as equivalent to the set B (symbolized
by A ≡ B) if and only if there exists a third set the members of which are ordered
pairs such that: (1) the first member of each pair is an element of A and the
second is an element of B, and (2) each member of A occurs as a first member and
each member of B occurs as a second member of exactly one pair. Thus,
if A and B are finite and A ≡ B, then the third set that establishes this fact provides
a pairing, or matching, of the elements of A with those of B. Conversely, if it is
possible to match the elements of A with those of B, thenA ≡ B, because a set of
pairs meeting requirements (1) and (2) can be formed—i.e., ifa ∊ A is matched
with b ∊ B, then the ordered pair (a, b) is one member of the set. By thus defining
equivalence of sets in terms of the notion of matching, equivalence is formulated
independently of finiteness. As an illustration involving infinite sets, Nmay be
taken to denote the set of natural numbers 0, 1, 2, … (some authors exclude 0
from the natural numbers). Then {(n, n
2
) | n ∊ N} establishes the seemingly
paradoxical equivalence of N and the subset of N formed by the squares of the
natural numbers.
As stated previously, a set B is included in, or is a subset of, a set A (symbolized
byB ⊆ A) if every element of B is an element of A. So defined, a subset may

79. possibly include all of the elements of A, so that A can be a subset of itself.
Furthermore, the empty set, because it by definition has no elements that are not
included in other sets, is a subset of every set.
If every element of set B is an element of set A, but the converse is false
(henceB ≠ A), then B is said to be properly included in, or is a proper subset
of, A(symbolized by B ⊂ A). Thus, if A = {3, 1, 0, 4, 2}, both {0, 1, 2} and
{0, 1, 2, 3, 4} are subsets of A; but {0, 1, 2, 3, 4} is not a proper subset. A finite set
is nonequivalent to each of its proper subsets. This is not so, however, for infinite
sets, as is illustrated with the set N in the earlier example. (The equivalence
of N and its proper subset formed by the squares of its elements was noted
by Galileo Galilei in 1638, who concluded that the notions of less than, equal to,
and greater than did not apply to infinite sets.)

CARDINALITY AND TRANSFINITE NUMBERS

The application of the notion of equivalence to infinite sets was first
systematically explored by Cantor. With N defined as the set of natural numbers,
Cantor’s initial significant finding was that the set of all rational numbers is
equivalent to N but that the set of all real numbers is not equivalent to N. The
existence of nonequivalent infinite sets justified Cantor’s introduction of
“transfinite” cardinal numbers as measures of size for such sets. Cantor defined
the cardinal of an arbitrary set A as the concept that can be abstracted
from A taken together with the totality of other equivalent sets. Gottlob Frege, in
1884, and Bertrand Russell, in 1902, both mathematical logicians, defined the
cardinal number of a set A somewhat more explicitly, as the set of all sets that
are equivalent to A. This definition thus provides a place for cardinal numbers as
objects of a universe whose only members are sets.
The above definitions are consistent with the usage of natural numbers as
cardinal numbers. Intuitively, a cardinal number, whether finite (i.e., a natural
number) or transfinite (i.e., nonfinite), is a measure of the size of a set. Exactly
how a cardinal number is defined is unimportant; what is important is that if
and only if A ≡ B.

80. To compare cardinal numbers, an ordering relation (symbolized by <) may be
introduced by means of the definition if A is equivalent to a subset
of B and B is equivalent to no subset of A. Clearly, this relation is irreflexive
and transitive: and imply.
When applied to natural numbers used as cardinals, the relation < (less than)
coincides with the familiar ordering relation for N, so that < is an extension of that
relation.
The symbol ℵ0 (aleph-null) is standard for the cardinal number of N (sets of this
cardinality are called denumerable), and ℵ (aleph) is sometimes used for that of
the set of real numbers. Then n < ℵ0 for each n ∊ N and ℵ0 < ℵ.
This, however, is not the end of the matter. If the power set of a set A—
symbolizedP(A)—is defined as the set of all subsets of A, then, as Cantor
proved, for every set A—a relation that is known as Cantor’s theorem. It
implies an unending hierarchy of transfinite cardinals: .
Cantor proved thatand suggested that there are no cardinal numbers
between ℵ0 and ℵ, a conjecture known as the continuum hypothesis.
There is an arithmetic for cardinal numbers based on natural definitions
of addition,multiplication, and exponentiation (squaring, cubing, and so on), but
this arithmetic deviates from that of the natural numbers when transfinite
cardinals are involved. For example, ℵ0 + ℵ0 = ℵ0 (because the set of integers is
equivalent to N), ℵ0 · ℵ0 = ℵ0 (because the set of ordered pairs of natural numbers
is equivalent to N), and c + ℵ0 = c for every transfinite cardinal c (because every
infinite set includes a subset equivalent to N).
The so-called Cantor paradox, discovered by Cantor himself in 1899, is the
following. By the unrestricted principle of abstraction, the formula “x is a set”
defines a set U; i.e., it is the set of all sets. Now P(U) is a set of sets and so P(U) is a
subset of U. By the definition of < for cardinals, however, if A ⊆ B, then it is not
the case that . Hence, by substitution,. But by Cantor’s theorem,
. This is a contradiction. In 1901 Russell devised another contradiction of
a less technical nature that is now known as Russell’s paradox. The formula “x is a
set and (x ∉ x)” defines a set R of all sets not members of themselves.
Using proof by contradiction, however, it is easily shown that (1) R ∊ R. But then

81. by the definition of R it follows that (2) (R ∉ R). Together, (1) and (2) form a
contradiction.

Set theory


Written by: Robert R. Stoll
Axiomatic set theory

In contrast to naive set theory, the attitude adopted in an axiomatic development
of set theory is that it is not necessary to know what the “things” are that are
called “sets” or what the relation of membership means. Of sole concern are the
properties assumed about sets and the membership relation. Thus, in an
axiomatic theory of sets, set and the membership relation ∊ are undefined terms.
The assumptions adopted about these notions are called the axioms of the
theory. Axiomatic set theorems are the axioms together with statements that can
be deduced from the axioms using the rules of inference provided by a system of
logic. Criteria for the choice of axioms include: (1) consistency—it should be
impossible to derive as theorems both a statement and its negation; (2)
plausibility—axioms should be in accord with intuitive beliefs about sets; and (3)
richness—desirable results of Cantorian set theory can be derived as theorems.

The Zermelo-Fraenkel axioms
The first axiomatization of set theory was given in 1908 by Ernst Zermelo, a
German mathematician. From his analysis of the paradoxes described above in
the sectionCardinality and transfinite numbers, he concluded that they are
associated with sets that are “too big,” such as the set of all sets in Cantor’s
paradox. Thus, the axioms that Zermelo formulated are restrictive insofar as the
asserting or implying of the existence of sets is concerned. As a consequence,

82. there is no apparent way, in his system, to derive the known contradictions from
them. On the other hand, the results of classical set theory short of the paradoxes
can be derived. Zermelo’s axiomatic theory is here discussed in a form that
incorporates modifications and improvements suggested by later
mathematicians, principally Thoralf Albert Skolem, a Norwegian pioneer
in metalogic, and Abraham Adolf Fraenkel, an Israeli mathematician. In the
literature on set theory, it is called Zermelo-Fraenkel set theory and abbreviated
ZFC (“C” because of the inclusion of the axiom of choice).

SCHEMAS FOR GENERATING WELL-FORMED FORMULAS

The ZFC “axiom of extension” conveys the idea that, as in naive set theory, a set is
determined solely by its members. It should be noted that this is not merely a
logically necessary property of equality but an assumption about the membership
relation as well.
The set defined by the “axiom of the empty set” is the empty (or null) set Ø.
For an understanding of the “axiom schema of separation”
considerable explanationis required. Zermelo’s original system included the
assumption that, if a formula S(x) is “definite” for all elements of a set A, then
there exists a set the elements of which are precisely those elements x of A for
which S(x) holds. This is a restricted version of the principle of abstraction, now

83. known as the principle of comprehension, for it provides for the existence of sets
corresponding to formulas. It restricts that principle, however, in two ways: (1)
Instead of asserting the existence of sets unconditionally, it can be applied only in
conjunction with preexisting sets, and (2) only “definite” formulas may be used.
Zermelo offered only a vague description of “definite,” but clarification was given
by Skolem (1922) by way of a precise definition of what will be called simply a
formula of ZFC. Using tools of modern logic, the definition may be made as
follows:
 I. For any variables x and y, x ∊ y and x = y are formulas (such formulas are
called atomic).
 II. If S and T are formulas and x is any variable, then each of the following is a
formula: If S, then T; S if and only if T; S and T; S or T; not S; for all x, S; for
some x,T.
Formulas are constructed recursively (in a finite number of systematic steps)
beginning with the (atomic) formulas of (I) and proceeding via the constructions
permitted in (II). “Not (x ∊ y),” for example, is a formula (which is abbreviated
tox ∉ y), and “There exists an x such that for every y, y ∉ x” is a formula. A
variable isfree in a formula if it occurs at least once in the formula without being
introduced by one of the phrases “for some x” or “for all x.” Henceforth, a
formula S in which xoccurs as a free variable will be called “a condition on x” and
symbolized S(x). The formula “For every y, x ∊ y,” for example, is a condition on x.
It is to be understood that a formula is a formal expression—i.e.,
a term without meaning. Indeed, a computer can be programmed to generate
atomic formulas and build up from them other formulas of ever-increasing
complexity using logical connectives (“not,” “and,” etc.) and operators (“for all”
and “for some”). A formula acquires meaning only when an interpretation of the
theory is specified; i.e., when (1) a nonempty collection (called the domain of the
interpretation) is specified as the range of values of the variables (thus the term
set is assigned a meaning, viz., an object in the domain), (2) the membership
relation is defined for these sets, (3) the logical connectives and operators are
interpreted as in everyday language, and (4) the logical relation of equality is
taken to be identity among the objects in the domain.

84. The phrase “a condition on x” for a formula in which x is free is merely suggestive;
relative to an interpretation, such a formula does impose a condition on x. Thus,
the intuitive interpretation of the “axiom schema of separation” is: given a
set A and a condition on x, S(x), those elements of A for which the condition holds
form a set. It provides for the existence of sets by separating off certain elements
of existing sets. Calling this the axiom schema of separation is appropriate,
because it is actually a schema for generating axioms—one for each choice of S(x).

AXIOMS FOR COMPOUNDING SETS

Although the axiom schema of separation has a constructive quality, further
means of constructing sets from existing sets must be introduced if some of the
desirable features of Cantorian set theory are to be established. Three axioms—
axiom of pairing, axiom of union, and axiom of power set—are of this sort.
By using five of the axioms (2–6), a variety of basic concepts of naive set theory
(e.g., the operations of union, intersection, and Cartesian product; the notions of
relation, equivalence relation, ordering relation, and function) can be defined
with ZFC. Further, the standard results about these concepts that were attainable
in naive set theory can be proved as theorems of ZFC.
AXIOMS FOR INFINITE AND ORDERED SETS

If I is an interpretation of an axiomatic theory of sets, the sentence that results
from an axiom when a meaning has been assigned to “set” and “∊,” as specified
by I, is either true or false. If each axiom is true for I, then I is called a model of the
theory. If the domain of a model is infinite, this fact does not imply that any
object of the domain is an “infinite set.” An infinite set in the latter sense is an
object d of the domain D of I for which there is an infinity of distinct objects d′
in D such that d′Edholds (E standing for the interpretation of ∊). Though the
domain of any model of the theory of which the axioms thus far discussed are
axioms is clearly infinite, models in which every set is finite have been devised.
For the full development of classical set theory, including the theories of real

85. numbers and of infinite cardinal numbers, the existence of infinite sets is needed;
thus the “axiom of infinity” is included.
The existence of a unique minimal set ω having properties expressed in the axiom
of infinity can be proved; its distinct members are Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø,
{Ø}}}, … . These elements are denoted by 0, 1, 2, 3, … and are called natural
numbers. Justification for this terminology rests with the fact that the Peano
postulates (five axioms published in 1889 by the Italian mathematician Giuseppe
Peano), which can serve as a base for arithmetic, can be proved as theorems in
set theory. Thereby the way is paved for the construction within ZFC of entities
that have all the expected properties of the real numbers.
The origin of the axiom of choice was Cantor’s recognition of the importance of
being able to “well-order” arbitrary sets—i.e., to define an ordering relation for a
given set such that each nonempty subset has a least element. The virtue of a
well-ordering for a set is that it offers a means of proving that a property holds for
each of its elements by a process (transfinite induction) similar to mathematical
induction. Zermelo (1904) gave the first proof that any set can be well-ordered.
His proof employed a set-theoretic principle that he called the “axiom of choice,”
which, shortly thereafter, was shown to be equivalent to the so-called well-
orderingtheorem.
Intuitively, the axiom of choice asserts the possibility of making a simultaneous
choice of an element in every nonempty member of any set; this guarantee
accounts for its name. The assumption is significant only when the set has
infinitely many members. Zermelo was the first to state explicitly the axiom,
although it had been used but essentially unnoticed earlier (see also Zorn’s
lemma). It soon became the subject of vigorous controversy because of its
nonconstructive nature. Some mathematicians rejected it totally on this ground.
Others accepted it but avoided its use whenever possible. Some changed their
minds about it when its equivalence with the well-ordering theorem was proved
as well as the assertion that any two cardinal numbers c and d are comparable
(i.e., that exactly one of c < d, d < c, c = d holds). There are many other equivalent
statements, though even today a few mathematicians feel that the use of the
axiom of choice is improper. To the vast majority, however, it, or an equivalent
assertion, has become an indispensable and commonplace tool. (Because of this

86. controversy, ZFC was adopted as an acronym for the majority position with the
axiom of choice and ZF for the minority position without the axiom of choice.)

SCHEMA FOR TRANSFINITE INDUCTION AND
ORDINAL ARITHMETIC

When Zermelo’s axioms were found to be inadequate for a full-blown
development of transfinite induction and ordinal arithmetic, Fraenkel and Skolem
independently proposed an additional axiom schema to eliminate the difficulty.
As modified byJohn von Neumann, a Hungarian-born American mathematician, it
says, intuitively, that if with each element of a set there is associated exactly one
set, then the collection of the associated sets is itself a set; i.e., it offers a way to
“collect” existing sets to form sets. As an illustration, each of ω, P(ω), P(P(ω)), … ,
formed by recursively taking power sets (set formed of all the subsets of the
preceding set), is a set in the theory based on Zermelo’s original eight axioms. But
there appears to be no way to establish the existence of the set having all these
sets as its members. However, an instance of the “axiom schema of replacement”
provides for its existence.
Intuitively, the axiom schema of replacement is the assertion that, if the domain
of a function is a set, then so is its range. That this is a powerful schema (in
respect to the further inferences that it yields) is suggested by the fact that the
axiom schema of separation can be derived from it and that, when applied in
conjunction with the axiom of power set, the axiom of pairing can be deduced.
The axiom schema of replacement has played a significant role in developing a
theory of ordinal numbers. In contrast to cardinal numbers, which serve to
designate the size of a set, ordinal numbers are used to determine positions
within a prescribed well-ordered sequence. Under an approach conceived by von
Neumann, if A is a set, the successor A′ of A is the set obtained by adjoining A to
the elements of A (A′ = A ∪ {A}). In terms of this notion the natural numbers, as
defined above, are simply the succession 0, 0′, 0″, 0‴, … ; i.e., the natural numbers
are the sets obtained starting with Ø and iterating the prime operation a finite
number of times. The natural numbers are well-ordered by the ∊ relation, and
with this ordering they constitute the finite ordinal numbers. The axiom of infinity

87. secures the existence of the set of natural numbers, and the set ω is the first
infinite ordinal. Greater ordinal numbers are obtained by iterating the prime
operation beginning with ω. An instance of the axiom schema of replacement
asserts that ω, ω′, ω″, … form a set. The union of this set and ω is the still greater
ordinal that is denoted by ω2 (employing notation from ordinal arithmetic). A
repetition of this process beginning with ω2 yields the ordinals (ω2)′, (ω2)″, … ;
next after all of those of this form is ω3. In this way the sequence of ordinals ω,
ω2, ω3, … is generated. An application of the axiom schema of replacement then
yields the ordinal that follows all of these in the same sense in which ω follows
the finite ordinals; in the notation from ordinal arithmetic, it is ω
2
. At
this point the iteration process can be repeated. In summary, the axiom schema
of replacement together with the other axioms make possible the extension of
the counting process as far beyond the natural numbers as one chooses.
In the ZFC system, cardinal numbers are defined as certain ordinals. From the
well-ordering theorem (a consequence of the axiom of choice), it follows that
every set Ais equivalent to some ordinal number. Also, the totality of ordinals
equivalent to Acan be shown to form a set. Then a natural choice for the cardinal
number of A is the least ordinal to which A is equivalent. This is the motivation for
defining a cardinal number as an ordinal that is not equivalent to any smaller
ordinal. The arithmetics of both cardinal and ordinal numbers have been fully
developed. That of finite cardinals and ordinals coincides with the arithmetic of
the natural numbers. For infinite cardinals, the arithmetic is uninteresting since,
as a consequence of the axiom of choice, both the sum and product of two such
cardinals are equal to the maximum of the two. In contrast, the arithmetic of
infinite ordinals is interesting and presents a wide assortment of oddities.
In addition to the guidelines already mentioned for the choice of axioms of ZFC,
another guideline is taken into account by some set theorists. For the purposes of
foundational studies of mathematics, it is assumed that mathematics is
consistent; otherwise, any foundation would fail. It may thus be reasoned that, if
a precise account of the intuitive usages of sets by mathematicians is given, an
adequate and correct foundation will result. Traditionally, mathematicians deal
with the integers, with real numbers, and with functions. Thus, an
intuitive hierarchy of sets in which these entities appear should be a model of

88. ZFC. It is possible to construct such a hierarchy explicitly from the empty set by
iterating the operations of forming power sets and unions in the following way.
The bottom of the hierarchy is composed of the sets A0 = Ø, A1, … , An, … , in
which each An + 1 is the power set of the preceding An. Then one can form the
union Aω of all sets constructed thus far. This can be followed by iterating the
power set operation as before: Aω′ is the power set of Aω and so forth. This
construction can be extended to arbitrarily high transfinite levels. There is no
highest level of the hierarchy; at each level, the union of what has been
constructed thus far can be taken and the power set operation applied to the
elements. In general, for each ordinal number α one obtains a set Aα, each
member of which is a subset of some Aβthat is lower in the hierarchy. The
hierarchy obtained in this way is called the iterative hierarchy. The domain of the
intuitive model of ZFC is conceived as the union of all sets in the iterative
hierarchy. In other words, a set is in the model if it is an element of some set Aα of
the iterative hierarchy.

AXIOM FOR ELIMINATING INFINITE DESCENDING SPECIES

From the assumptions that this system of set theory is sufficiently comprehensive
for mathematics and that it is the model to be “captured” by the axioms of ZFC, it
may be argued that models of axioms that differ sharply from this system should
be ruled out. The discovery of such a model led to the formulation by von
Neumann of axiom 10, the axiom of restriction, or foundation axiom.
This axiom eliminates from the models of the first nine axioms those in which
there exist infinite descending ∊-chains (i.e., sequences x1, x2, x3, … such
that x2 ∊ x1, x3 ∊x2, …), a phenomenon that does not appear in the model based on
an iterative hierarchy described above. (The existence of models having such
chains was discovered by the Russian mathematician Dimitry Mirimanoff in 1917.)
It also has other attractive consequences; e.g., a simpler definition of the notion
of ordinal number is possible. Yet there is no unanimity among mathematicians
whether there are sufficient grounds for adopting it as an additional axiom. On
the one hand, the axiom is equivalent (in a theory that allows only sets) to the
statement that every set appears in the iterative hierarchy informally described

89. above—there are no other sets. So it formulates the view that this is what the
universe of all sets is really like. On the other hand, there is no compelling need to
rule out sets that might lie outside the hierarchy—the axiom has not been shown
to have any mathematical applications.

The Neumann-Bernays-Gödel axioms

The second axiomatization of set theory originated with John von Neumann in the
1920s. His formulation differed considerably from ZFC because the notion of
function, rather than that of set, was taken as undefined, or “primitive.” In a
series of papers beginning in 1937, however, the Swiss logician Paul Bernays, a
collaborator with the German formalist David Hilbert, modified the von Neumann
approach in a way that put it in much closer contact with ZFC. In 1940, the
Austrian-born American logician Kurt Gödel, known for his undecidability proof,
further simplified the theory. This axiomatic version of set theory is called NBG,
after the Neumann-Bernays-Gödel axioms. As will be explained shortly, NBG is
closely related to ZFC, but it allows explicit treatment of so-called classes:
collections that might be too large to be sets, such as the class of all sets or the
class of all ordinal numbers.

90. For expository purposes it is convenient to adopt two undefined notions for
NBG:class and the binary relation ∊ of membership (though, as is also true in ZFC,
∊ suffices). For the intended interpretation, variables take classes—the totalities
corresponding to certain properties—as values. A class is defined to be a set if it is
a member of some class; those classes that are not sets are called proper classes.
Intuitively, sets are intended to be those classes that are adequate for
mathematics, and proper classes are thought of as those collections that are “so
big” that, if they were permitted to be sets, contradictions would follow. In NBG,
the classical paradoxes are avoided by proving in each case that the collection on
which theparadox is based is a proper class—i.e., is not a set.
Comments about the axioms that follow are limited to features that distinguish
them from their counterpart in ZFC. The axiom schema for class formation is
presented in a form to facilitate a comparison with the axiom schema of
separation of ZFC. In a detailed development of NBG, however, there appears
instead a list of seven axioms (not schemas) that state that, for each of certain
conditions, there exists a corresponding class of all those sets satisfying the

91. condition. From this finite set of axioms, each an instance of the above schema,
the schema (in a generalized form) can be obtained as a theorem. When obtained
in this way, the axiom schema for class formation of NBG is called the class
existence theorem.
In brief, axioms 4 through 8 of NBG are axioms of set existence. The same is true
of the next axiom, which for technical reasons is usually phrased in a more
general form. Finally, there may appear in a formulation of NBG an analog of the
last axiom of ZFC (axiom of restriction).
A comparison of the two theories that have been formulated is in order. In
contrast to the axiom schema of replacement of ZFC, the NBG version is not an
axiom schema but an axiom. Thus, with the comments above about the ZFC
axiom schema of separation in mind, it follows that NBG has only a finite number
of axioms. On the other hand, since the axiom schema of replacement of ZFC
provides an axiom for each formula, ZFC has infinitely many axioms—which is
unavoidable because it is known that no finite subset yields the full system of
axioms. The finiteness of the axioms for NBG makes the logical study of the
system simpler. The relationship between the theories may be summarized by the
statement that ZFC is essentially the part of NBG that refers only to sets. Indeed,
it has been proved that every theorem of ZFC is a theorem of NBG and that any
theorem of NBG that speaks only about sets is a theorem of ZFC. From this it
follows that ZFC is consistent if and only if NBG is consistent.
Limitations of axiomatic set theory
The fact that NBG avoids the classical paradoxes and that there is no apparent
way to derive any one of them in ZFC does not settle the question of
the consistency of either theory. One method for establishing the consistency of
an axiomatic theory is to give a model—i.e., an interpretation of the undefined
terms in another theory such that the axioms become theorems of the other
theory. If this other theory is consistent, then that under investigation must be
consistent. Such consistency proofs are thus relative: the theory for which a
model is given is consistent if that from which the model is taken is consistent.
The method of models, however, offers no hope for proving the consistency of an

92. axiomatic theory of sets. In the case of set theory and, indeed, of axiomatic
theories generally, the alternative is a direct approach to the problem.
If T is the theory of which the (absolute) consistency is under investigation, this
alternative means that the proposition “There is no sentence of T such that both
it and its negation are theorems of T” must be proved. The mathematical theory
(developed by the formalists) to cope with proofs about an axiomatic theory T is
called proof theory, or metamathematics. It is premised upon the formulation
of T as a formal axiomatic theory—i.e., the theory of inference (as well as T) must
be axiomatized. It is then possible to present T in a purely symbolic form—i.e., as
a formal language based on an alphabet the symbols of which are those for the
undefined terms of T and those for the logical operators and connectives. A
sentence in this language is a formula composed from the alphabet according to
prescribed rules. The hope for metamathematics was that, by using only
intuitively convincing, weak number-theoretic arguments (called finitary
methods), unimpeachable proofs of the consistency of such theories as axiomatic
set theory could be given.
That hope suffered a severe blow in 1931 from a theorem proved by Kurt Gödel
about any formal theory S that includes the usual vocabulary of elementary
arithmetic. By coding the formulas of such a theory with natural numbers (now
called Gödel numbers) and by talking about these numbers, Gödel was able to
make the metamathematics of S become part of the arithmetic of S and hence
expressible in S. The theorem in question asserts that the formula of S that
expresses (via a coding) “S is consistent” in S is unprovable in S if S is consistent.
Thus, if S is consistent, then the consistency of S cannot be proved within S;
rather, methods beyond those that can be expressed or reflected in S must be
employed. Because, in both ZFC and NBG, elementary arithmetic can be
developed, Gödel’s theorem applies to these two theories. Although there
remains the theoretical possibility of a finitary proof of consistency that cannot be
reflected in the foregoing systems of set theory, no hopeful, positive results have
been obtained.
Other theorems of Gödel when applied to ZFC (and there are corresponding
results for NBG) assert that, if the system is consistent, then (1) it contains a
sentence such that neither it nor its negation is provable (such a sentence is

93. called undecidable), (2) there is no algorithm (or iterative process) for deciding
whether a sentence of ZFC is a theorem, and (3) these same statements hold for
any consistent theory resulting from ZFC by the adjunction of further axioms or
axiom schemas. Apparently ZFC can serve as a foundation for all of present-day
mathematics because every mathematical theorem can be translated into and
proved within ZFC or within extensions obtained by adding suitable axioms. Thus,
the existence of undecidable sentences in each such theory points out an
inevitable gap between the sentences that are true in mathematics and sentences
that are provable within a single axiomatic theory. The fact that there is more to
conceivable mathematics than can be captured by the axiomatic approach
prompted the American logician Emil Post to comment in 1944 that
“mathematical thinking is, and must remain, essentially creative.”

Present status of axiomatic set theory
The foundations of axiomatic set theory are in a state of significant change as a
result of new discoveries. The situation with alternate (and conflicting) axiom
systems for set theory is analogous to the 19th-century revolution
in geometry that was set off by the discovery of non-Euclidean geometries. It is
difficult to predict the ultimate consequences of these late 20th-century findings
for set theory, but already they have had profound effects on attitudes about
certain axioms and have forced the realization of a continuous search for
additional axioms. These discoveries have focused attention on the concept of the
independence of an axiom. If T is an axiomatic theory and S is a sentence (i.e., a
formula) of T that is not an axiom, and ifT + S denotes the theory that results
from T upon the adjunction of S to T as a further axiom, then S is said to be
consistent with T if T + S is consistent and independent of T whenever both S and
∼S (the negation of S) are consistent with T. Thus, if S is independent of T, then
the addition of S or ∼S to T yields a consistent theory. The role of the axiom of
restriction (AR) can be clarified in terms of the notion of independence. If ZF′
denotes the theory obtained from ZF by deleting AR and either retaining or
deleting the axiom of choice (AC), then it can be proved that, if ZF′ is consistent,
AR is independent of ZF′.

94. Of far greater significance for the foundations of set theory is the status of AC
relative to the other axioms of ZF. The status in ZF of the continuum
hypothesis (CH) and its extension, the generalized continuum hypothesis (GCH),
are also of profound importance. In the following discussion of these questions,
ZF denotes Zermelo-Fraenkel set theory without AC. The first finding was
obtained by Kurt Gödel in 1939. He proved that AC and GCH are consistent
relative to ZF (i.e., if ZF is consistent, then so is ZF + AC + GCH) by showing that a
contradiction within ZF + AC + GCH can be transformed into a contradiction in ZF.
In 1963 American mathematician Paul Cohen proved that (1) if ZF is consistent,
then so is ZF + AC + ∼CH, and (2) if ZF is consistent, then so is ZF + ∼AC. Since in
ZF + AC it can be demonstrated that GCH implies CH, Gödel’s theorem together
with Cohen’s establishes the independence of AC and CH. For his proofs Cohen
introduced a new method (called forcing) of constructing interpretations of
ZF + AC. The method of forcing is applicable to many problems in set theory, and
since 1963 it has been used to give independence proofs for a wide variety of
highly technical propositions. Some of these results have opened new avenues for
attacks on important foundational questions.
The current unsettled state of axiomatic set theory can be sensed by the
responses that have been made to the question of how to regard CH in the light
of its independence from ZF + AC. Someone who believes that set theory deals
only with nonexistent fictions will have no concern about the question. But for
most mathematicians sets actually exist; in particular, ω and P(ω) exist (the set of
the natural numbers and its power set, respectively). Further, it should be the
case that every nondenumerable subset of P(ω) either is or is not equivalent
to P(ω); i.e., CH either is true or is false. Followers of this faith regard the axioms
of set theory as describing some well-defined reality—one in which CH must be
either true or false. Thus there is the inescapable conclusion that the present
axioms do not provide a complete description of that reality. A search for new
axioms is in progress. One who hopes to prove CH as a theorem must look for
axioms that restrict the number of sets. There seems to be little hope for this
restriction, however, without changing the intuitive notion of the set. Thus the
expectations favour the view that CH will be disproved. This disproof requires an
axiom that guarantees the existence of more sets—e.g., of sets having

95. cardinalities greater than those that can be proved to exist in ZF + AC. So far, none
of the axioms that have been proposed that are aimed in this direction (called
“generalized axioms of infinity”) serves to prove ∼CH. Although there is little
supporting evidence, the optimists hope that the status of thecontinuum
hypothesis will eventually be settled.

The Implicit Definition of the Set-Concept
 F. A. Muller

Abstract
Once Hilbert asserted that the axioms of a theory `define` theprimitive concepts
of its language `implicitly'. Thus whensomeone inquires about the meaning of the
set-concept, thestandard response reads that axiomatic set-theory defines
itimplicitly and that is the end of it. But can we explainthis assertion in a manner
that meets minimum standards ofphilosophical scrutiny? Is Jané (2001) wrong
when hesays that implicit definability is ``an obscure notion''? Doesan explanation
of it presuppose any particular view on meaning?Is it not a scandal of the
philosophy of mathematics that no answersto these questions are around? We
submit affirmative answers to allquestions. We argue that a Wittgensteinian
conception of meaninglooms large beneath Hilbert's conception of implicit
definability.Within the specific framework of Horwich's recent
Wittgensteiniantheory of meaning called semantic deflationism, we explain
anexplicit conception of implicit definability, and then go on toargue that, indeed,
set-theory, defines the set-conceptimplicitly according to this conception. We also
defend Horwich'sconception against a recent objection from the Neo-Fregeans
Hale and Wright (2001). Further, we employ the philosophicalresources gathered
to dissolve all traditional worries about thecoherence of the set-concept, raisedby
Frege, Russell and Max Black, and whichrecently have been defended vigorously
by Hallett (1984) in hismagisterial monographCantorian set-theory and
limitationof size. Until this day, scandalously, these worries havebeen ignored too
by philosophers of mathematics.

96. Introduction to Sets
Forget everything you know about numbers.
In fact, forget you even know what a number is.
This is where mathematics starts.
Instead of math with numbers, we will now think about math with "things".
Definition
What is a set? Well, simply put, it's a collection.
First we specify a common property among "things" (this word will be defined
later) and then we gather up all the "things" that have this common property.

For example, the items you wear: shoes, socks, hat, shirt, pants, and so on.
I'm sure you could come up with at least a hundred.
This is known as a set.

97. Or another example is types of fingers.
This set includes index, middle, ring, and pinky.


So it is just things grouped together with a certain property in common.
Notation
There is a fairly simple notation for sets. We simply list each element (or
"member") separated by a comma, and then put some curly brackets around
the whole thing:

The curly brackets { } are sometimes called "set brackets" or "braces".
This is the notation for the two previous examples:
{socks, shoes, watches, shirts, ...}
{index, middle, ring, pinky}
Notice how the first example has the "..." (three dots together).
The three dots ... are called an ellipsis, and mean "continue on".
So that means the first example continues on ... for infinity.

98. (OK, there isn't really an infinite amount of things you could wear, but I'm not
entirely sure about that! After an hour of thinking of different things, I'm still
not sure. So let's just say it is infinite for this example.)
So:
 The first set {socks, shoes, watches, shirts, ...} we call an infinite set,
 the second set {index, middle, ring, pinky} we call a finite set.
But sometimes the "..." can be used in the middle to save writing long lists:
Example: the set of letters:
{a, b, c, ..., x, y, z}
In this case it is a finite set (there are only 26 letters, right?)
Numerical Sets
So what does this have to do with mathematics? When we define a set, all we
have to specify is a common characteristic. Who says we can't do so with
numbers?
Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
Set of odd numbers: {..., -3, -1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}
And the list goes on. We can come up with all different types of sets.
There can also be sets of numbers that have no common property, they are
just defined that way. For example:

99. {2, 3, 6, 828, 3839, 8827}
{4, 5, 6, 10, 21}
{2, 949, 48282, 42882959, 119484203}
Are all sets that I just randomly banged on my keyboard to produce.
Why are Sets Important?
Sets are the fundamental property of mathematics. Now as a word of warning,
sets, by themselves, seem pretty pointless. But it's only when we apply sets in
different situations do they become the powerful building block of mathematics
that they are.
Math can get amazingly complicated quite fast. Graph Theory, Abstract
Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number
Theory, and the list goes on. But there is one thing that all of these share
in common: Sets.
Universal Set


At the start we used the word "things" in quotes. We call this
the universal set. It's a set that contains everything. Well,
not exactly everything. Everything that is relevant to our
question.


Then our sets included integers. The universal set for that would
be all the integers. In fact, when doing Number Theory, this is
almost always what the universal set is, as Number Theory is
simply the study of integers.


However in Calculus (also known as real analysis), the universal
set is almost always the real numbers. And in complex analysis,
you guessed it, the universal set is the complex numbers.

100. Some More Notation

When talking about sets, it is fairly standard to use Capital Letters to represent
the set, and lowercase letters to represent an element in that set.

So for example, A is a set, and a is an element in A. Same with B and b, and C and
c.
Now you don't have to listen to the standard, you can use something like m to
represent a set without breaking any mathematical laws (watch out, you can
get π years in math jail for dividing by 0), but this notation is pretty nice and
easy to follow, so why not?
Also, when we say an element a is in a set A, we use the symbol to show it.
And if something is not in a set use .
Example: Set A is {1,2,3}. We can see that 1 A, but 5 A
Equality
Two sets are equal if they have precisely the same members. Now, at first
glance they may not seem equal, so we may have to examine them closely!
Example: Are A and B equal where:
 A is the set whose members are the first four positive whole numbers
 B = {4, 2, 1, 3}
Let's check. They both contain 1. They both contain 2. And 3, And 4. And we
have checked every element of both sets, so: Yes, they are equal!
And the equals sign (=) is used to show equality, so we write:
A = B

101. Subsets
When we define a set, if we take pieces of that set, we can form what is called
a subset.
So for example, we have the set {1, 2, 3, 4, 5}. A subset of this is {1, 2, 3}.
Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset,
since it contains an element (6) which is not in the parent set. In general:
A is a subset of B if and only if every element of A is in B.
So let's use this definition in some examples.
Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?
1 is in A, and 1 is in B as well. So far so good.
3 is in A and 3 is also in B.
4 is in A, and 4 is in B.
That's all the elements of A, and every single one is in B, so we're done.
Yes, A is a subset of B
Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we
only look at the elements in A.
Let's try a harder example.
Example: Let A be all multiples of 4 and B be all multiples of 2. Is A a subset of B?
And is B a subset of A?
Well, we can't check every element in these sets, because they have an infinite
number of elements. So we need to get an idea of what the elements look like
in each, and then compare them.

102. The sets are:
 A = {..., -8, -4, 0, 4, 8, ...}
 B = {..., -8, -6, -4, -2, 0, 2, 4, 6, 8, ...}
By pairing off members of the two sets, we can see that every member of A is
also a member of B, but every member of B is not a member of A:


So:
A is a subset of B, but B is not a subset of A
Proper Subsets
If we look at the defintion of subsets and let our mind wander a bit, we come to
a weird conclusion.
Let A be a set. Is every element in A an element in A? (Yes, I wrote that
correctly.)
Well, umm, yes of course, right?
So doesn't that mean that A is a subset of A?
This doesn't seem very proper, does it? We want our subsets to be proper. So
we introduce (what else but) proper subsets.

103. A is a proper subset of B if and only if every element in A is also in B, and
there exists at least one element in B that is not in A.
This little piece at the end is only there to make sure that A is not a proper
subset of itself. Otherwise, a proper subset is exactly the same as a normal
subset.
Example:
{1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.
Example:
{1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in
the first set.
Notice that if A is a proper subset of B, then it is also a subset of B.
Even More Notation
When we say that A is a subset of B, we write A B.
Or we can say that A is not a subset of B by A B ("A is not a subset of B")
When we talk about proper subsets, we take out the line underneath and so it
becomes A B or if we want to say the opposite, A B.
Empty (or Null) Set
This is probably the weirdest thing about sets.

104. As an example, think of the set of piano keys on a guitar.
"But wait!" you say, "There are no piano keys on a guitar!"
And right you are. It is a set with no elements.
This is known as the Empty Set (or Null Set).There aren't any elements in it.
Not one. Zero.
It is represented by
Or by {} (a set with no elements)
Some other examples of the empty set are the set of countries south of the
south pole.
So what's so weird about the empty set? Well, that part comes next.
Empty Set and Subsets
So let's go back to our definition of subsets. We have a set A. We won't define it
any more than that, it could be any set. Is the empty set a subset of A?
Going back to our definition of subsets, if every element in the empty set is
also in A, then the empty set is a subset of A . But what if we
have no elements?

105. It takes an introduction to logic to understand this, but this statement is one
that is "vacuously" or "trivially" true.
A good way to think about it is: we can't find any elements in the empty set
that aren't in A, so it must be that all elements in the empty set are in A.
So the answer to the posed question is a resounding yes.
The empty set is a subset of every set, including the empty set itself.
Order
No, not the order of the elements. In sets it does not matter what order the
elements are in.
Example: {1,2,3,4} is the same set as {3,1,4,2}
When we say "order" in sets we mean the size of the set.
Just as there are finite and infinite sets, each has finite and infinite order.
For finite sets, we represent the order by a number, the number of elements.
Example, {10, 20, 30, 40} has an order of 4.
For infinite sets, all we can say is that the order is infinite. Oddly enough, we
can say with sets that some infinities are larger than others, but this is a more
advanced topic in sets.
Arg! Not more notation!
Nah, just kidding. No more notation.

106. by
Ricky Shadrach
set definition

A set is a group or collection of objects or numbers, considered as an entity
unto itself. Sets are usually symbolized by uppercase, italicized, boldface
letters such as A, B, S, or Z. Each object or number in a set is called a
member or element of the set. Examples include the set of all computers in
the world, the set of all apples on a tree, and the set of all irrational numbers
between 0 and 1.
When the elements of a set can be listed or denumerated, it is customary to
enclose the list in curly brackets. Thus, for example, we might speak of the set
(call it K) of all natural numbers between, and including, 5 and 10 as:
K = {5, 6, 7, 8, 9, 10}
A set can have any non-negative quantity of elements, ranging from none (the
empty set or null set) to infinitely many. The number of elements in a set is
called the cardinality, and can range from zero to denumerably infinite (for the
sets of natural numbers, integers, or rational numbers) to non-denumerably
infinite for the sets of irrational numbers, real numbers, imaginary numbers, or
complex numbers).

107. The most basic relations in set theory can be summarized as follows.
 A set S1 is a subset of set S if and only if every element of S1 is also an
element of S.
 A set S1 is a proper subset of set S if and only if every element of S1 is also
an element of S, but there are some elements in S that are not elements of
S1.
 The intersection of two sets S and T is the set X of all elements x such that
x is in S and x is in T.
 The union of two sets S and T is the set Y of all elements y such that y is in
S or y is in T, or both.
Relationships between and among sets can be illustrated by means of a
special type of drawing called a Venn diagram. The table below denotes
common set symbology.

Set theory is fundamental to all of mathematics. In its "pure" form, set theory
can be esoteric and even bizarre, and is primarily of interest to academics.
However, set theory is closely connected with symbolic logic, and these fields

108. are becoming increasingly relevant in software engineering, especially in the
fields of artificial intelligence and communications security.

What are the different types of sets?
The different types of sets are explained below with examples.
Empty Set or Null Set:
A set which does not contain any element is called an empty set, or the null
set or the void set and it is denoted by ∅ and is read as phi. In roster form, ∅
is denoted by {}. An empty set is a finite set, since the number of elements
in an empty set is finite, i.e., 0.
For example: (a) The set of whole numbers less than 0.

(b) Clearly there is no whole number less than 0.

Therefore, it is an empty set.

(c) N = {x : x ∈ N, 3 < x < 4}

• Let A = {x : 2 < x < 3, x is a natural number}

Here A is an empty set because there is no natural number between
2 and 3.

• Let B = {x : x is a composite number less than 4}.

Here B is an empty set because there is no composite number less than 4.

Note:
∅ ≠ {0} ∴ has no element.

{0} is a set which has one element 0.

The cardinal number of an empty set, i.e., n(∅) = 0


Singleton Set:

109. A set which contains only one element is called a singleton set.

For example:
• A = {x : x is neither prime nor composite}

It is a singleton set containing one element, i.e., 1.

• B = {x : x is a whole number, x < 1}

This set contains only one element 0 and is a singleton set.

• Let A = {x : x ∈ N and x² = 4}

Here A is a singleton set because there is only one element 2 whose square
is 4.

• Let B = {x : x is a even prime number}

Here B is a singleton set because there is only one prime number which is
even, i.e., 2.


Finite Set:
A set which contains a definite number of elements is called a finite set.
Empty set is also called a finite set.

For example:
• The set of all colors in the rainbow.

• N = {x : x ∈ N, x < 7}

• P = {2, 3, 5, 7, 11, 13, 17, ...... 97}


Infinite Set:
The set whose elements cannot be listed, i.e., set containing never-ending
elements is called an infinite set.

For example:

110. • Set of all points in a plane

• A = {x : x ∈ N, x > 1}

• Set of all prime numbers

• B = {x : x ∈ W, x = 2n}

Note:
All infinite sets cannot be expressed in roster form.

For example:
The set of real numbers since the elements of this set do not follow any
particular pattern.



Cardinal Number of a Set:
The number of distinct elements in a given set A is called the cardinal
number of A. It is denoted by n(A).

For example:
• A {x : x ∈ N, x < 5}

A = {1, 2, 3, 4}

Therefore, n(A) = 4

• B = set of letters in the word ALGEBRA

B = {A, L, G, E, B, R}

Therefore, n(B) = 6


Equivalent Sets:
Two sets A and B are said to be equivalent if their cardinal number is same,
i.e., n(A) = n(B). The symbol for denoting an equivalent set is ‘↔’.

For example:

111. A = {1, 2, 3} Here n(A) = 3

B = {p, q, r} Here n(B) = 3

Therefore, A ↔ B

Equal sets:
Two sets A and B are said to be equal if they contain the same elements.
Every element of A is an element of B and every element of B is an element
of A.

For example:
A = {p, q, r, s}

B = {p, s, r, q}

Therefore, A = B


The various types of sets and their definitions are explained above with the
help of examples.
● Set Theory
● Sets
● Objects Form a Set
● Elements of a Set
● Properties of Sets
● Representation of a Set
● Different Notations in Sets
● Standard Sets of Numbers
● Types of Sets
● Pairs of Sets
● Subset
● Subsets of a Given Set

112. ● Operations on Sets
● Union of Sets
● Intersection of Sets
● Difference of two Sets
● Complement of a Set
● Cardinal number of a set
● Cardinal Properties of Sets
● Venn Diagrams

Types of Sets
1. Various types of sets:
 Finite set
A set which contains limited number of elements is called a finite set.
Example1. A = {1, 3, 5, 7, 9}.
Here A is a set of five positive odd numbers less than 10. Since the number of
elements is limited, A is a finite set.
2. A grade 5 class is a finite set, as the number of students is a fixed number.
 Infinite set
A set which contains unlimited number of elements is called an infinite set.
1. The set of natural numbers N, is an infinite set as the counting of numbers does
not come to an end.
2. The set of integers is an infinite set.

113. Singleton set
A set which contains only one element is a singleton set.
Example 1:
A {set of even prime numbers}
Now A = {2}.
The only even prime number is 2. All other prime numbers are odd.
Therefore A can contain only one element, namely 2.
Therefore A is a singleton set.
Example 2:
X = {x: x is neither positive nor negative}
Now, X = {0}, because it’s only 0 which is neither positive nor negative.
Therefore, X is a singleton set.
 Null set
A Set which does not contain any element is called empty set or null set.
S = {x: x ∈Z, x = 1/n, n ∈ N}
N is natural number and Z is integer.
Since n is an integer, 1/n cannot be an integer. Therefore, S cannot contain an
element x which is an integer.
Note:
1. The Empty set is denoted as { } or by the greek letter Φ
2. {{}} or {Φ} are not empty sets, because each contain one element, namely the
empty set Φ itself.
 Cardinal Number of a set or Cardinality of a set:

114. The cardinality of a set is the number of elements a set contains. It is denoted as n
(A).
n (A) is read as the number of elements in set A
Example 1:
A = {1, 2, 3, 4, 5}
The cardinality of set A is 5.
It is denoted as n (A) = 5
Note:
Example 1.
Let X = { }, then n (X) = 0
Let Z = {{}} or Z = {Φ}, then n (Y) = 1
Example 2:
Cardinality of infinite set is not defined.
 Equivalent sets
Two sets which have the same number of elements, i.e. same cardinality are
equivalent sets.
P = {p. q. r, s, t} and Q = {a, e, i, o, u}
Since the two sets P and Q contain the same number of elements 5, therefore
they are equivalent sets.
 Equal sets
Two sets that contain the same elements are called equal sets.
A = {1, 2, 3, 4, 5, 6, 7, 8, 9} and B = {
 Overlapping sets

115. Two sets that have at least one common element are called overlapping sets.
Example 1:
X = {1, 2, 3} and Y = {3, 4, 5}
The two sets X and Y have an element 3 in common. Therefore they are called
overlapping sets.
Example 2:
A = {x: x is an even prime number}
B = {x: x∈2n,n∈N }
The two sets A and B are overlapping sets because 2 is a common element in A
and B.
 Disjoint sets
C = {2, 4, 6} and D = {1, 3, 5}
The two sets C and D are disjoint sets as they do not have even one element in
common.
Example 1:
E = {set of odd numbers} and F = {set of even numbers}
The two sets E and F are disjoint as no odd number is an even number nor any of
even numbers is odd.
Subset
Set A is a subset of set B if every element of A is an element of set B.
If set A is a subset of set B, then it is denoted as ACB
Example 1:
Let A = {1, 2, 3} and B = {2, 3, 4, 1}
Since every element of set A is present in set B too, we say A is a subset of B.

116. Example 2:
Let A = {multiples of 4} B = {multiples of 2}
Now, X = {1, 4, 8, 12...} and Y = {1, 2, 4, 6, 8, 10, 12...}
Since every element of set X is also an element of set Y, therefore
X is a subset of Y.
Note:
1. If two sets A and B are equal sets, then each one is a subset of the other.
If A = {a, e, i, o, u} and B = {vowels of English alphabets}, then A = B.
But, note that A C B and BCA.
Therefore, if A C B and BCA, then A = B
2. Every set is a subset of itself.
ACA
3. empty set is a subset of every set.

117. Sets & Basic Operations
Sets and Elements
A set is a well-defined collection of objects, each of which is called
an element or member of the set.
Sets can either be defined by (i) listing out its members (such as the set
of vowels a, e, i, o, u), or (ii) by stating out rules or properties (such as
the set of the names of the capital cities of Indian states).
Notations
A set is normally denoted by a capital letter, such as A, X etc., while the
elements might be denoted generically by lowercase letters like a, b etc.
The elements of a set are enclosed within curly braces '{' and '}', and are
separated from each other by commas.
Sets can be specified in two ways:
1. Tabular form:
The elements are listed explicitly. e.g., .
2. Set-builder notation or property method:
The set is defined by stating the properties which characterize the
elements of the set. e.g., B={x:x is an odd integer, x>0}
The above reads as “ is a set of  such that  is an odd integer
and ”. x stands for any element of the set, the colon is read as
“such that” and the comma as “and.” In explicit form, the above set
is .
The symbolic way of writing the statement that an element “ is an
element of set ” or, equivalently, “ belongs to set ,” isp∈A
When specifying two elements, like  and , we writep,q∈A

118. The statement “ does not belong to ” is written asa∉A
The set  of vowels can be defined as:A={x:x is a letter of
the English alphabet, and x is a vowel}
In the above set, note that a,u∈A, while b∉A
Two sets  and  are equal if both have the same elements. This is
denoted as . If they areunequal, it is expressed as A≠B
Note that changing the arrangement of the elements of a set does not change
the set. Also, repetition of elements of a set does not change the set.
Three sets are defined as follows:
,  and 
, since both contain the same elements, even though in different
order. , since both contain the same elements, even though  has the
element  repeated.
Universal Set and Empty Set
The universal set, or the universe, is a larger set assumed to contain
all sets under consideration for a particular study. The universal set is
denoted by . Thus, for a study on car models, the universal set could
consist of all the car models in the world ever made.
An empty set, or a null set, is a set with no elements. It is denoted
by ∅ or . Evidently, there can be only one null set.
Subsets
If every element of a set  is also an element of set , then  is called
a subset of . This is also expressed as:  is contained in ,
or  contains . This is expressed symbolically asA⊆B

119. Inversely, B is called a superset of . Symbolically, this is expressed
asB⊇A
The statement “A is not a subset of B” implies that at least one element
of A does not belong to B. This is expressed as A⊆B or B⊇A.
From the definition of subset, if ,  is still a subset of , and vice
versa.
Three sets are defined as follows:
,  and 
We can see that C⊆A and C⊆B, as the elements  and  of  are also elements
of  and . However, A⊆B, since some elements of  are not elements of .
Based on what we have learned till now, some properties of sets are
1. Every set  is a subset of the universal set , since by definition, every
element of a set belongs to .
2. The empty set ∅ is a subset of any set .
3. Every set A is a subset of itself. Thus, A⊆A
4. Two set are equal if and only if both are subsets of each other. In other
words,  if and only if A⊆B and B⊆A
5. If every element of a set  belongs to a set , and every element of
set  belongs to a set , then clearly every element of set  belongs to
set . Thus, if A⊆B and B⊆C, then A⊆C.
Proper Subset
If A⊆B and A≠B, then  is a proper subset of . This is expressed
asA⊂B
Thus, all proper subsets are subsets, but not all subsets are proper
subsets.

120. Disjoints sets

Fig 1: Venn diagram denoting disjoint
sets  and 
If two sets have no elements in common, they are said to be disjoint.
Two sets which are disjoint can never have a superset-subset
relationship, unless one is a null set.
Venn Diagrams
A venn diagram is a pictorial representation of sets, wherein they are
shown as enclosed areas. Typically, the universal set  is represented
by the area within a rectangle, and the other sets as circles placed within
the rectangle.

Fig 2: Venn diagram depicting  as
subset of : A⊆B
Fig. 1 represents a universal set and disjoint sets  and , while
Fig. 2represents a universal set with sets  and  having subset-
superset relationship.

121. Set Operations
The three common operations on sets are the operations of union,
intersection and difference.
Union

Fig 3: A∪B
The union of two sets,  and , denoted by A∪B, is the set of all
elements which belong to either  or ; i.e.
A∪B={x:x∈A  or  x∈B}
The venn diagram depicting the union relationship is shown in Fig. 3.
Intersection

Fig 4: A∩B
The intersection of two sets,  and , denoted by A∩B, is the set of all
elements which belong to both  and ; i.e.
A∩B={x:x∈A  and  x∈B}

122. The venn diagram depicting the union relationship is shown in Fig. 4.
Three sets are defined as follows:
,  and 
A∪B={1,2,4,6,7,8,10,12} A∩B={4,10}
B∪C={1,2,3,4,5,6,7,8,10,12} B∩C=∅
A∪C={1,3,4,5,7,10} A∩C={1,7}
Let  denote the set of students in a class. Let set  and
set  denote the collection of boys and girls respectively in the
class.
We have,
B∪G=U, as each student in  is either in set  or set .
B∩G=∅, as there is no student belonging to both sets  and .
Properties of unions and intersections
Some of the properties of the union and intersections of sets are as
follows:
1. (A∩B)⊆A and (A∩B)⊆B, since every element of A∩B belongs to , as
well as . This is also clear from the venn diagram of Fig. 4.
2. A⊆(A∪B) and B⊆(A∪B), since every element of  belongs to A∪B, and
similarly every element of  belongs to A∪B. This is also clear from the
venn diagram of Fig. 3.
Combining the above two properties leads to:(A∩B)⊆A⊆(A∪B) and
(A∩B)⊆B⊆(A∪B)

123. Fig 5: The complement of set , is
shown shaded
Complement
The complement, or absolute complement, of a set , denoted
by

or , is the set of all elements which belong to  (the universal
set) but does not belong to .A'={x:x∈U  and  x∉A}(see Fig. 5)

Fig 6: A~B
Difference
The difference of two sets, A and B, denoted by , or , is the
set of all elements which belong to  but do not belong to ,
i.e.A~B={x:x∈A, x∉B}(see Fig. 6)
http://www.virtualnerd.com/tutorials/?id=Alg1_9_2_1

124. Definition of
Ordered Pair
more ...

Two numbers written in a certain order.
Usually written in parentheses like this: (4,5)

Can be used to show the position on a graph, where the "x" (horizontal) value is
first, and the "y" (vertical) value is second.

Here the point (12,5) is 12 units along, and 5 units up.
ordered pair

An ordered pair is a pair of numbers used to locate a point on a coordinate plane; the
first number tells how far to move horizontally and the second number tells how far to
move vertically.

Venn Diagram

125. A schematic diagram used in logic theory to depict collections of sets and
represent their relationships.
The Venn diagrams on two and three sets are illustrated above. The order-
two diagram (left) consists of two intersecting circles, producing a total of
four regions, , , , and (the empty set, represented by none of the
regions occupied). Here, denotes the intersection of sets and .
The order-three diagram (right) consists of three symmetrically placed
mutually intersecting circles comprising a total of eight regions. The regions
labeled , , and consist of members which are only in one set and no
others, the three regions labelled , , and consist of members
which are in two sets but not the third, the region consists of
members which are simultaneously in all three, and no regions occupied
represents .
In general, an order- Venn diagram is a collection of simple closed
curves in the plane such that
1. The curves partition the plane into connected regions, and
2. Each subset of corresponds to a unique region formed by the
intersection of the interiors of the curves in (Ruskey).
Since there are (the binomial coefficient) ways to pick members from
a total of , the number of regions in an order Venn diagram is

126. (where the region outside the diagram is included in the count).
The region of intersection of the three circles in the order three
Venn diagram in the special case of the center of each being located at
theintersection of the other two is a geometric shape known as a Reuleaux
triangle.

The left figure at left above shows an Venn diagram due to Branko
Grünbaum, while the attractive 7-fold rosette illustrated in the middle figure
is an Venn diagram called "Victoria" by Ruskey. The right figure shows
a recently constructed symmetric Venn diagram on due to Ruskey,
Carla Savage, and Stan Wagon.
In Season 4 episode "Power" of the television crime drama NUMB3RS,
mathematical genius Charles Eppes constructs a Venn diagram to
determine suspects who match a particular description and have a history
of violence.
http://www.purplemath.com/modules/venndiag.htm

127. Sets and Venn Diagrams
Sets

A set is a collection of things.
For example, the items you wear is a set: these would include shoes, socks,
hat, shirt, pants, and so on.
You write sets inside curly brackets like this:
{socks, shoes, pants, watches, shirts, ...}
You can also have sets of numbers:
 Set of whole numbers: {0, 1, 2, 3, ...}
 Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Ten Best Friends
You could have a set made up of your ten best friends:
 {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}
Each friend is an "element" (or "member") of the set (it is normal to
use lowercase letters for them.)

128. Now let's say that alex, casey, drew and hunter play Soccer:
Soccer = {alex, casey, drew, hunter}
(The Set "Soccer" is made up of the elements alex, casey, drew and hunter).

And casey, drew and jade play Tennis:
Tennis = {casey, drew, jade}
You could put their names in two separate circles:

Union
You can now list your friends that play Soccer OR Tennis.

129. This is called a "Union" of sets and has the special symbol ∪:
Soccer ∪ Tennis = {alex, casey, drew, hunter, jade}
Not everyone is in that set ... only your friends that play Soccer or Tennis (or
both).
We can also put it in a "Venn Diagram":

Venn Diagram: Union of 2 Sets
A Venn Diagram is clever because it shows lots of information:
 Do you see that alex, casey, drew and hunter are in the "Soccer" set?
 And that casey, drew and jade are in the "Tennis" set?
 And here is the clever thing: casey and drew are in BOTH sets!
Intersection
"Intersection" is when you have to be in BOTH sets.
In our case that means they play both Soccer AND Tennis ... which is casey
and drew.
The special symbol for Intersection is an upside down "U" like this: ∩
And this is how we write it down:

130. Soccer ∩ Tennis = {casey, drew}
In a Venn Diagram:

Venn Diagram: Intersection of 2 Sets

Which Way Does That "U" Go?

Think of them as "cups": ∪ would hold more water than ∩, right?
So Union ∪ is the one with more elements than Intersection ∩
Difference
You can also "subtract" one set from another.
For example, taking Soccer and subtracting Tennis means people that play
Soccer but NOT Tennis ... which is alex and hunter.
And this is how we write it down:

131. Soccer − Tennis = {alex, hunter}
In a Venn Diagram:

Venn Diagram: Difference of 2 Sets
Summary So Far
 ∪ is Union: is in either set
 ∩ is Intersection: must be in both sets
 − is Difference: in one set but not the other
Three Sets
You can also use Venn Diagrams for 3 sets.
Let us say the third set is "Volleyball", which drew, glen and jade play:
Volleyball = {drew, glen, jade}
But let's be more "mathematical" and use a Capital Letter for each set:
 S means the set of Soccer players
 T means the set of Tennis players
 V means the set of Volleyball players
The Venn Diagram is now like this:

132. Union of 3 Sets: S ∪ T ∪ V
You can see (for example) that:
 drew plays Soccer, Tennis and Volleyball
 jade plays Tennis and Volleyball
 alex and hunter play Soccer, but don't play Tennis or Volleyball
 no-one plays only Tennis
We can now have some fun with Unions and Intersections ...

This is just the set S
S = {alex, casey, drew, hunter}

133. This is the Union of Sets T and V
T ∪ V = {casey, drew, jade, glen}


This is the Intersection of Sets S and V
S ∩ V = {drew}
And how about this ...
 take the previous set S ∩ V
 then subtract T:

134. This is the Intersection of Sets S and V minus Set T
(S ∩ V) − T = {}
Hey, there is nothing there!
That is OK, it is just the "Empty Set". It is still a set, so we use the curly
brackets with nothing inside: {}
The Empty Set has no elements: {}
Universal Set
The Universal Set is the set that contains everything. Well,
not exactlyeverything. Everything that we are interested in now.
Sadly, the symbol is the letter "U" ... which is easy to confuse with the ∪ for
Union. You just have to be careful, OK?
In our case the Universal Set is our Ten Best Friends.
U = {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}
We can show the Universal Set in a Venn Diagram by putting a box around the
whole thing:

135. Now you can see ALL your ten best friends, neatly sorted into what sport they
play (or not!).
And then we can do interesting things like take the whole set and subtract the
ones who play Soccer :

We write it this way:
U − S = {blair, erin, francis, glen, ira, jade}
Which says "The Universal Set minus the Soccer Set is the Set {blair, erin,
francis, glen, ira, jade}"
In other words "everyone who does not play Soccer".

136. Complement
And there is a special way of saying "everything that is not", and it is
called "complement" .
We show it by writing a little "C" like this:
S
c

Which means "everything that is NOT in S", like this:

S
c
= {blair, erin, francis, glen, ira, jade}
(just like the U − C example from above)

Summary
 ∪ is Union: is in either set
 ∩ is Intersection: must be in both sets
 − is Difference: in one set but not the other
 A
c
is the Complement of A: everything that is not in A

137.  Empty Set: the set with no elements. Shown by {}
 Universal Set: all things we are interested in
DEFINITION
Venn diagram
A Venn diagram is an illustration of the relationships between and among sets,
groups of objects that share something in common. Usually, Venn diagrams are
used to depict setintersections (denoted by an upside-down letter U). This type of
diagram is used in scientific and engineering presentations, in theoretical
mathematics, in computer applications, and in statistics.
The drawing is an example of a Venn diagram that shows the relationship among
three overlapping sets X, Y, and Z. The intersection relation is defined as the
equivalent of the logic AND. An element is a member of the intersection of two
sets if and only if that element is a member of both sets. Venn diagrams are
generally drawn within a large rectangle that denotes the universe, the set of all
elements under consideration.

In this example, points that belong to none of the sets X, Y, or Z are gray. Points
belonging only to set X are cyan in color; points belonging only to set Y are
magenta; points belonging only to set Z are yellow. Points belonging to X and Y
but not to Z are blue; points belonging to Y and Z but not to X are red; points

138. belonging to X and Z but not to Y are green. Points contained in all three sets are
black.
Here is a practical example of how a Venn diagram can illustrate a situation. Let
the universe be the set of all computers in the world. Let X represent the set of all
notebook computers in the world. Let Y represent the set of all computers in the
world that are connected to the Internet. Let Z represent the set of all computers
in the world that have anti-virus software installed. If you have a notebook
computer and surf the Net, but you are not worried about viruses, your computer
is probably represented by a point in the blue region. If you get concerned about
computer viruses and install an anti-virus program, the point representing your
computer will move into the black area.
Cartesian Product
The Cartesian product of two sets and (also called the product set, set
direct product, or cross product) is defined to be the set of all points
where and . It is denoted , and is called the Cartesian product
since it originated in Descartes' formulation of analytic geometry. In the
Cartesian view, points in the plane are specified by their vertical and
horizontal coordinates, with points on a line being specified by just one
coordinate. The main examples of direct products are Euclidean three-
space ( , where are the real numbers), and the plane ().
The graph product is sometimes called the Cartesian product (Vizing 1963,
Clark and Suen 2000).

Graph Cartesian Product

139. The Cartesian graph product , sometimes simply called "the"
graph product (Beineke and Wilson 2004, p. 104) of graphs and with
disjoint point sets and and edge sets and is the graph with point
set and adjacent with whenever
or (Harary 1994, p. 22).
Graph Cartesian products can be computed in the Wolfram
Language using GraphComputation`GraphProduct [G1, G2, "Cartes
ian"] orGraphProduct[G1, G2] in the Wolfram
Language package Combinatorica` .
The following table gives examples of some graph Cartesian products.
Here denotes a path graph and a cycle graph.
product result
grid graph

ladder graph
grid graph

prism graph
prism graph

book graph
stacked book graph

140. hypercube graph

141. Chapter 5:
Functions

142. What is a Function?
A function relates an input to an output.

It is like a machine that has an input and an output.
And the output is related somehow to the input.

f(x)
"f(x) = ... " is the classic way of writing a function.
And there are other ways, as you will see!
Input, Relationship, Output
We will see many ways to think about functions, but there are always three
main parts:
 The input
 The relationship
 The output
Example: "Multiply by 2" is a very simple function.
Here are the three parts:
Input Relationship Output
0 × 2 0
1 × 2 2

143. 7 × 2 14
10 × 2 20
... ... ...
For an input of 50, what is the output?
Some Examples of Functions
 x
2
(squaring) is a function
 x
3
+1 is also a function
 Sine, Cosine and Tangent are functions used in trigonometry
 and there are lots more!
But we are not going to look at specific functions ...
... instead we will look at the general idea of a function.
Names
First, it is useful to give a function a name.
The most common name is " f", but we can have other names like "g" ... or even
"marmalade" if we want.
But let's use "f":

144. We say "f of x equals x squared"
what goes into the function is put inside parentheses () after the name of the
function:
So f(x) shows us the function is called "f", and "x" goes in
And we usually see what a function does with the input:
f(x) = x
2
shows us that function "f" takes "x" and squares it.

Example: with f(x) = x
2
:
 an input of 4
 becomes an output of 16.
In fact we can write f(4) = 16.

The "x" is Just a Place-Holder!
Don't get too concerned about "x", it is just there to show us where the input
goes and what happens to it.
It could be anything!
So this function:
f(x) = 1 - x + x
2

Is the same function as:
 f(q) = 1 - q + q
2

145.  h(A) = 1 - A + A
2

 w(θ) = 1 - θ + θ
2

The variable (x, q, A, etc) is just there so we know where to put the values:
f(2) = 1 - 2 + 2
2
= 3

Sometimes There is No Function Name
Sometimes a function has no name, and we see something like:
y = x
2

But there is still:
 an input (x)
 a relationship (squaring)
 and an output (y)
Relating
At the top we said that a function was like a machine. But a function doesn't
really have belts or cogs or any moving parts - and it doesn't actually destroy
what we put into it!
A function relates an input to an output.
Saying "f(4) = 16" is like saying 4 is somehow related to 16. Or 4 → 16

146. Example: this tree grows 20 cm every year, so the height of the tree
is related to its age using the function h:
h(age) = age × 20
So, if the age is 10 years, the height is:
h(10) = 10 × 20 = 200 cm
Here are some example values:
age h(age) = age × 20
0 0
1 20
3.2 64
15 300
... ...

What Types of Things Do Functions Process?
"Numbers" seems an obvious answer, but ...

147. ... which numbers?
For example, the tree-height function h(age) =
age×20 makes no sense for an age less than zero.

... it could also be letters ("A"→"B"), or ID codes
("A6309"→"Pass") or stranger things.
So we need something more powerful, and that is where sets come in:



A set is a collection of things.
Here are some examples:
Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
Set of clothes: {"hat","shirt",...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}
Each individual thing in the set (such as "4" or "hat") is called a member,
or element.
So, a function takes elements of a set, and gives back elements of a set.
A Function is Special
But a function has special rules:
 It must work for every possible input value
 And it has only one relationship for each input value

148. This can be said in one definition:

Formal Definition of a Function
A function relates each element of a set
with exactly one element of another set
(possibly the same set).
The Two Important Things!
1. "...each element..." means that every element in X is related to
some element in Y.
We say that the function covers X (relates every element of it).
(But some elements of Y might not be related to at all, which is fine.)
2. "...exactly one..." means that a function is single valued. It will
not give back 2 or more results for the same input.
So "f(2) = 7 or 9" is not right!
Note: "One-to-many" is not allowed, but "many-to-one" is allowed:

149. (one-to-many) (many-to-one)
This is NOT OK in a function But this is OK in a function
When a relationship does not follow those two rules then it is not a function ...
it is still arelationship, just not a function.
Example: The relationship x → x
2


Could also be written as a table:
X: x Y: x
2

3 9
1 1
0 0
4 16
-4 16

150. ... ...

It is a function, because:
 Every element in X is related to Y
 No element in X has two or more relationships
So it follows the rules.
(Notice how both 4 and -4 relate to 16, which is allowed.)
Example: This relationship is not a function:

It is a relationship, but it is not a function, for these reasons:
 Value "3" in X has no relation in Y
 Value "4" in X has no relation in Y
 Value "5" is related to more than one value in Y
(But the fact that "6" in Y has no relationship does not matter)

151. Vertical Line Test
On a graph, the idea of single
valued means that no vertical line ever
crosses more than one value.
If it crosses more than once it is still a
valid curve, but isnot a function.


Some types of functions have stricter rules, to find out more you can
read Injective, Surjective and Bijective
Infinitely Many
My examples have just a few values, but functions usually work on sets with
infinitely many elements.
Example: y = x
3

 The input set "X" is all Real Numbers
 The output set "Y" is also all the Real Numbers
We can't show ALL the values, so here are just a few examples:
X: x Y: x
3

-2 -8
-0.1 -0.001

152. 0 0
1.1 1.331
3 27
and so on... and so on...

Domain, Codomain and Range
In our examples above
 the set "X" is called the Domain,
 the set "Y" is called the Codomain, and
 the set of elements that get pointed to in Y (the actual values produced by
the function) is called the Range.
We have a special page on Domain, Range and Codomain if you want to know
more.
So Many Names!
Functions have been used in mathematics for a very long time, and lots of
different names and ways of writing functions have come about.
Here are some common terms you should get familiar with:

153. Example: with z = 2u
3
:
 "u" could be called the "independent variable"
 "z" could be called the "dependent variable" (it depends on the value of u)
Example: with f(4) = 16:
 "4" could be called the "argument"
 "16" could be called the "value of the function"
Ordered Pairs
And here is another way to think about functions:
Write the input and output of a function as an "ordered pair", such as
(4,16).
They are called ordered pairs because the input always comes first, and the
output second:
(input, output)
So it looks like this:
( x, f(x) )

154. Example:
(4,16) means that the function takes in "4" and gives out "16"
Set of Ordered Pairs
A function can then be defined as a set of ordered pairs:
Example: {(2,4), (3,5), (7,3)} is a function that says
"2 is related to 4", "3 is related to 5" and "7 is related 3".
Also, notice that:
 the domain is {2,3,7} (the input values)
 and the range is {4,5,3} (the output values)
But the function has to be single valued, so we also say
"if it contains (a, b) and (a, c), then b must equal c"
Which is just a way of saying that an input of "a" cannot produce two different
results.
Example: {(2,4), (2,5), (7,3)} is not a function because {2,4} and {2,5}
means that 2 could be related to 4 or 5.
In other words it is not a function because it is not single valued

155. A Benefit of Ordered Pairs
We can graph them...
... because they are also coordinates!
So a set of coordinates is also a function (if they follow
the rules above, that is)

A Function Can be in Pieces
We can create functions that behave differently depending on the input value
Example: A function with two pieces:
 when x is less than 0, it gives 5,
 when x is 0 or more it gives x
2


Here are some example values:
x y
-3 5
-1 5
0 0
2 4

156. 4 16
... ...

Read more at Piecewise Functions.
Explicit vs Implicit
One last topic: the terms "explicit" and "implicit".
"Explicit" is when the function shows us how to go directly from x to y, such as:
y = x
3
- 3
When we know x, we can find y
That is the classic y = f(x) style.
"Implicit" is when it is not given directly such as:
x
2
- 3xy + y
3
= 0
When we know x, how do we find y?
It may be hard (or impossible!) to go directly from x to y.
"Implicit" comes from "implied", in other words shown indirectly.

Definition of
Function
more ...

157. A function is a special relationship where each input has a single output.

It is often written as "f(x)" where x is the input value.

Example: f(x) = x/2 ("f of x is x divided by 2") is a function, because each input "x"
has a single output "x/2":
• f(2) = 1
• f(16) = 8
• f(−10) = −5

Definition of
Variable
more ...

A symbol for a number we don't know yet. It is usually a letter like x or y.

Example: in x + 2 = 6, x is the variable.
Definition of
Constant
more ...

158. A fixed value.

In Algebra, a constant is a number on its own, or sometimes a letter such as a, b or
c to stand for a fixed number.

Example: in "x + 5 = 9", 5 and 9 are constants

If it is not a constant it is called a variable.



Definition of
Parameter

A value that is already "built in" to a function.

159. Example: in the function h(year) = 20 × year
"year" is a variable and "20" is a parameter.


Parameters can be changed so that the function can be used for other
things.

Example: A different tree's growth rate is 30 cm per year, so its function
is h(year) = 30 × year

We could even make it more general by writing

h(age; rate) = rate × age

and in this case a semicolon (;) is used to separate the variable(s) from the
parameters(s)

Definition of
Domain of a function
more ...

All the values that go into a function

The output values are called the range.

160. Domain → Function → Range

Example: when the function f(x) = x
2
is given the values x = {1,2,3,...} then
{1,2,3,...} is the domain.
See: Range of a function

Domain and Range
When working with functions, we frequently come across two terms:
DOMAIN & RANGE. What is a domain? What is a range? Why are they
important?
Definition
Domain: The domain of a function is the set of all possible input values (often
the "x" variable), which produce a valid output from a particular function. It
is the set of all real numbers for which a function is mathematically defined.
Most often a simple function's domain is all real numbers. Consider a
simple linear equation like the graph shown below drawn from the function
y=.5x+10. What values are valid inputs? It's not a trick question -- every
real number! Its range is all real numbers because there is nothing you can
put in for x that won't work. That's why the graph extends forever in the x
directions (left and right).

161. What kind of functions don't have a domain of all real numbers? Well, if the
domain is the set of all real numbers for which the function is defined, then
logically we're looking for a function that has certain input values that do not
produce a valid output, i.e., the function is undefined for that input. Here is an
example:

This function is defined for almost any real x. But, what is the value of y when
x=1? Well, it's 3 divided by 0, which is undefined. Therefore 1 is not in the
domain of this function. All other real numbers are valid inputs, so the
domain is all real numbers except for x=1.
What other kinds of functions have domains that aren't all real numbers?
Certain "inverse" functions, like the inverse trig functions, have limited
domains as well. Since the sine function can only haveoutputs from -1 to 1, its
inverse can only accept inputs from -1 to 1. The domain of inverse sine is -1
to 1. However, the most common reason for limited domains is probably the divide by
zero issue. When finding the domain of a function, first look for any values
that cause you to divide by zero. Remember also that we cannot take the
square root of a negative number, so keep an eye out for situations where
the radicand (the "stuff" inside the square root sign) could result in a
negative value. In that case, it would not be a valid input so the domain
would not include such values.
Definition
Range: The range is the set of all possible output values (usually the variable
y, or sometimes expressed as f(x)), which result from using a particular
function.

162. The range of a simple linear function is almost always going to be all real
numbers. A graph of a line, such as the one shown below on the left, will
extend forever in either y direction. There's one notable exception:
y=constant. When you have a function where y equals a constant (like y=3),
your graph is a horizontal line. In that case, the range is just that one value.
Otherwise, the range is all real numbers.

Many other functions have limited ranges. While only a few types have
limited domains, you will frequenty see functions with unusual ranges. Here
are a few examples:

163. As you can see, these two functions have ranges that are limited. No matter
what values you enter into a sine function you will never get a result greater
than 1 or less than -1. No matter what values you enter into
you will never get a result less than -2.
Summary: The domain of a function is all the possible input values for which
the function is defined, and the range is all possible output values.

If you are still confused about domain and range, you might consider posting
your question on ourmessage board, or reading another website's lesson
on domain and range. Or, you can use the calculator below to determine the
domain and range of ANY equation:
Domain & Range Calculator
5.2 - Reference - Graphs of eight basic types of functions
The purpose of this reference section is to show you graphs of various types of
functions in order that you can become familiar with the types. You will discover that
each type has its own distinctive graph. By showing several graphs on one plot you
will be able to see their common features. Examples of the following types of
functions are shown in this gallery:
 linear
 quadratic
 power
 polynomial
 rational
 exponential
 logarithmic
 sinusoidal
In each case the argument (input) of the function is called x and the value (output) of
the function is called y.

164. Linear functions. These are functions
of the form:
y = m x + b,
where m and b are constants. A
typical use for linear functions is
converting from one quantity or set
of units to another. Graphs of these
functions arestraight lines. m is the
slope and b is the y intercept. If m is
positive then the line rises to the
right and if m is negative then the
line falls to the right. Linear functions
are described in detail here.




Quadratic functions. These are
functions of the form:
y = a x

2+ b x + c,
where a, b and c are constants. Their
graphs are called parabolas. This is
the next simplest type of function
after the linear function. Falling
objects move along parabolic paths.
If a is a positive number then the
parabola opens upward and if a is a

165. negative number then the parabola opens downward. Quadratic functions are
described in detail here.




Power functions. These are functions of the form:
y = a x
b
,

166. where a and b are constants. They get their name from the fact that the variable x is
raised to some power. Many physical laws (e.g. the gravitational force as a function
of distance between two objects, or the bending of a beam as a function of the load
on it) are in the form of power functions. We will assume that a = 1 and look at
several cases for b:

The power b is a positive integer. See
the graph to the right. When x = 0 these
functions are all zero. When x is big and
positive they are all big and positive.
When x is big and negative then the ones
with even powers are big and positive
while the ones with odd powers are
big and negative.


The power b is a negative integer. See the graph to the right. When x = 0 these
functions suffer a division by zero and therefore are all infinite. Whenx is big and
positive they are small and positive. When x is big and negative then the ones with

167. even powers are small and positive while the ones with odd powers are small and
negative.


The power b is a fraction between 0 and
1. See the graph to the right. When x = 0
these functions are all zero. The curves
are vertical at the origin and
as x increases they increase but curve
toward the x axis.


The power function is discussed in
detail here.




Polynomial functions. These are
functions of the form:
y = an · x
n
+ an −1 · x
n −1
+ …
+ a2 · x
2
+ a1 · x + a0,
where an, an −1, … , a2, a1, a0 are
constants. Only whole
number powers of x are allowed. The
highest power of x that occurs is
called the degree of the polynomial.
The graph shows examples of degree 4 and degree 5 polynomials. The degree gives
the maximum number of “ups and downs” that the polynomial can have and also the
maximum number of crossings of the x axis that it can have.

Polynomials are useful for generating smooth curves in computer graphics

168. applications and for approximating other types of functions. Polynomials are
described in detail here.




Rational functions. These
functions are the ratio of
two polynomials. One field
of study where they are
important is in stability
analysis of mechanical and
electrical systems (which
uses Laplace transforms).

When the polynomial in the
denominator is zero then the rational function
becomes infinite as indicated by a vertical
dotted line (called an asymptote) in its graph.
For the example to the right this happens
when x = −2 and when x = 7.

When x becomes very large the curve may
level off. The curve to the right levels off at y =
5.




The graph to the right shows another example of a rational function. This one has a
division by zero at x = 0. It doesn't level off but does approach the straight line y =

169. x when x is large, as indicated by the dotted line (another asymptote).




Exponential functions. These are
functions of the form:
y = a b
x
,
where x is in an exponent (not in the base
as was the case for power functions)
and a and b are constants. (Note that
only b is raised to the power x; not a.) If
the base b is greater than 1 then the result
is exponential growth. Many physical
quantities grow exponentially (e.g. animal
populations and cash in an interest-
bearing account).



If the base b is smaller than 1 then the
result is exponential decay. Many
quantities decay exponentially (e.g.
the sunlight reaching a given depth of
the ocean and the speed of an object
slowing down due to friction).


Exponential functions are described in
detail here.

170. Logarithmic functions. There
are many equivalent ways to
define logarithmic functions. We
will define them to be of the
form:
y = a ln (x) + b,
where x is in the natural
logarithm and a and b are
constants. They are only defined
for positive x. For small x they
are negative and for large xthey
are positive but stay small. Logarithmic functions accurately describe the response of
the human ear to sounds of varying loudness and the response of the human eye to
light of varying brightness. Logarithmic functions are described in detail here.




Sinusoidal functions. These are
functions of the form:
y = a sin (b x + c),
where a, b and c are constants.
Sinusoidal functions are useful
for describing anything that has
a wave shape with respect to
position or time. Examples are
waves on the water, the height of
the tide during the course of the
day and alternating current in
electricity. Parameter a (called

171. the amplitude) affects the height of the wave, b (the angular velocity) affects the
width of the wave and c (the phase angle) shifts the wave left or right. Sinusoidal
functions are described in detail here.








Types of Functions
Constant Function:
Let ‘A’ and ‘B’ be any two non–empty sets, then a function ‘f’ from ‘A’ to ‘B’ is
called Constant Function if and only if range of ‘f’ is a singleton.
Algebraic Function:
The function defined by algebraic expression are called algebraic function.
e.g. f(x)=x2+3x+6
Polynomial Function:
A function of the form P(x)=amxn+an−1xn−1+⋯+a1x+a0
Where ‘n’ is a positive integer and an,an−1,⋯,a1,a0are real number is called a
polynomial function of degree ‘n’.
Linear Function:
A polynomial function with degree ‘t’ is called a linear function. The most general
form of linear function is
f(x)=ax+b
Quadratic Function:
A polynomial function with degree ‘2’ is called a Quadratic function. The most
general form of Quadratic equation is f(x)=ax2+bx+c
Cubic Function:
A polynomial function with degree ‘3’ is called cubic function. The most general
form of cubic function is f(x)=ax3+bx2+cx+d
Identity Function:
Let f:A→Bbe a function then ‘f’ is called on identity function. If f(x)=x,∀x∈A.
Rational Function:
A function R(x) defined by R(x)=P(x)Q(x), where both P(x)andQ(x)are polynomial

172. function is called, rational function.
Trigonometric Function:
A functionf(x)=sinx, f(x)=cosxetc, then f(x)is called trigonometric function.
Exponential Function:
A function in which the variable appears as exponent (power) is called an
exponential function
e.g. (i) f(x)=ax (ii) f(x)=3x.
Logarithmic Function:
A function in which the variable appears as an argument of logarithmic is called
logarithmic function.
e.g. f(x)=loga(x).

173. Chapter 6:
Equations
&
Inequalities

174. Equation
more ...
An equation says that two things are equal. It will have an equals sign "=" like this:

7 + 2 = 10 − 1

That equation says: what is on the left (7 + 2) is equal to what is on the right (10
− 1)

So an equation is like a statement "this equals that"

Here is another equation:


equation

DefinitionA dd to F las hc ardsSave to F avoritesSee Examples
A mathematical statement used to evaluate a value. An equation can use
any combination of mathematical operations, including addition, subtraction, division,
or multiplication. An equation can be already established due to the properties of
numbers (2 + 2 = 4), or can be filled solely with variables which can be replaced with
numerical values to get a resulting value. For example, the equation to calculate return

175. on sales is: Net income ÷ Sales revenue = Return on Sales. When the values for net
income and sales revenue are plugged into the equation, you are able to calculate the
value of return on sales.

Use equation in a sentence
 People in the business world are using more math to come up with the
right equation to help out their bottom line.
20 people found this helpful
 In order to pass the first portion of the math class, which would determine whether
or not you'd graduate, you had to know your way around an equation.
17 people found this helpful
 The teacher took each of the individual grades and placed into an equation to
calculate each student's grade in the course.


R e a d more : http://www.businessdictionary.com/definition/equation.html#ixzz3zyd9FrTR
equation
Simple Definition of EQUATION
 mathematics : a statement that two expressions are equal (such as 8 + 3
= 11 or 2x – 3 = 7)
 : a complicated situation or issue
 : the act of regarding two things as the same : the act of equating
things





Full Definition of EQUATION
1. 1a : the act or process of equatingb (1) : an element affecting a
process : factor (2) : a complex of variable factorsc : a state of
being equated; specifically : a state of close association or

176. identification<bring governmental enterprises and payment for them
into immediate equation — R. G. Tugwell>
2. 2a : a usually formal statement of the equality or equivalence of
mathematical or logical expressionsb : an expression representing a
chemical reaction quantitatively by means of chemical symbols

Equation
An equation is a mathematical statement that two things are equal. It
consists of two expressions, one on each side of an 'equals' sign. For
example: This equation states that 12 is equal to the sum of 7 and
5, which is obviously true. In an equation, the left side is always equal to
the right side.

Types of Equations
1. Polynomial Equations
Polynomial equations are in the form P(x) = 0, where P(x) is a
polynomial.
Types of Polynomial Equations

177. 1.1 Linear Equations
Linear equations are equations of the type ax + b = 0,
with a ≠ 0, or any other equation in which the terms can be
operated and simplified into an equation of the same form.
(x + 1)
2
= x
2
- 2
x
2
+ 2x + 1 = x
2
- 2
2x + 1 = -2
2x + 3 = 0
1.2 Quadratic Equations
Quadratic equations are equations of the type ax
2
+ bx +
c = 0, with a ≠ 0.
Incomplete quadratic equations
ax
2
= 0
ax
2
+ bx = 0
ax
2
+ c = 0

178. 1.3 Cubic Equations
Cubic equations are equations of the type ax
3
+ bx
2
+ cx
+ d = 0, with a ≠ 0.
1.4 Quartic Equations
Quartic equations are equations of the type ax
4
+ bx
3
+
cx
2
+ dx + e = 0, with a ≠ 0.
Biquadratic Equations
Biquadratic equations are quartic equations that do not
have terms with an odd degree.
ax
4
+ bx
2
+ c = 0, with a ≠ 0.
2. Rational Polynomial Equations
The rational polynomial equations are of the
form , where P(x) and Q(x) are polynomials.

179. 3. Irrational Polynomial Equations
The irrational equations are those that have at least a
polynomial under the radical sign.



4. Transcendental Equations
The transcendental equations are equations that include
transcendental functions.
4.1 Exponential Equations
Exponential equations are equations in which the
unknown appears in the exponent.

180. 4.2 Logarithmic Equations
Logarithmic equations are equations in which the
unknown is affected by a logarithm.



4.3 Trigonometric Equations
Trigonometric equations are the equations in which the
unknown is affected by a trigonometric function.



Types of Equations

Algebra is a vast subject that studies about the algebraic expressions and
equations. An algebraic expression is the combination of constants and
variables; where, constants are the fixed quantities and variables are
quantities that susceptible to vary. Equations are found everywhere in

181. mathematics. An algebraic equation refers to an algebraic expression with
a symbol of equality (=). In other words, an equation is an expression which
has an equal to (=) sign between the two algebraic quantities or a set of
quantity.
For example - Few algebraic equations are
(1) 4m22n22 + 2m = 0
(2) 7a + 4b + 9c = - 8
(3) 2 sin A + cos A = 2cos22 A

Students may come across several different types of algebraic
equations :
(1) Linear equation
(2) Quadratic equation
(3) Polynomial equation
(4) Trigonometric equation
(5) Radical equation
(6) Exponential equation

Solving equations means to find the value or set of values of unknown
variable contained in it. Let us go ahead in this chapter and learn more
about different types of differential equations and sample problems based
on those.

Different Types of Equations
An equation is an important part of calculus, which comes under
mathematics. It helps us to solve many problem.
Types of Algebraic Equations
Lets discuss different types of algebraic equations:
1) Linear Equations:
A linear equation is an algebraic equation in which each term is either a
constant or the product of a constant and a single variable. The graph of
linear equation is a straight line if there are two variables.
General form of the linear equation with two variables:

182. y = mx + c, m ≠≠ 0.
Where, m is known as slope and c at that point on which it cut y axis.
Linear equations with different variables:
a) The equation with one variable:
An equation who have only one variable.
Examples:
1. 8a - 8 = 0
2. 9a = 72.
b) The equation with two variables:
An equation who have only two types of variable in the equation.
Examples:
1. 7x + 7y = 12
2. 8a - 8d = 74
3. 9a + 6b - 82 = 0.
c) The equation with three variables:
An equation who have only three types of variable in the equation.
Examples:
1. 5x + 7y - 6z = 12
2. 13a - 8b + 31c = 74
3. 6p + 14q -7r + 82 = 0.
2) Radical Equations:
An equation whose maximum exponent on the variable is 1212 and have
more than one term or we can say that a radical equation is an equation in
which the variable is lying inside a radical symbol usually in a square root.
Examples:

183. 1. x√+10=26x+10=26
2. x2−5−−−−−√+x−1x2−5+x−1
3) Quadratic Equations:
Quadratic equation is the second degree equation in one variable contains
the variable with an exponent of 2.
The general form:
ax
2
+ bx + c = 0, a ≠≠ 0
Examples of Quadratic Equations:
1. 8x
2
+ 7x - 75 = 0
2. 4y
2
+ 14y - 8 =0
3. 5a
2
- 5a = 35
4) Exponential Equations:
An equation who have variables in the place of exponents. Exponential
equation can be solved using the property: axax = ayay => x = y.
Examples:
1. a
b
= 0 Here "a" is base and "b" is exponent.
2. 4
x
=0
3. 8
x
= 32.
5) Rational Equations:
A rational equation is one that involves rational expressions.
Example:
x2x2 = x+24x+24.
Types of Math Equations
Here, we will solve some algebraic equations:

184. Solved Examples
Question 1: Solve the quadratic equation x
2
+ 6x - 27 = 0.
Solution:
Given quadratic equation: x
2
+ 6x - 27 = 0

x
2
+ 6x - 27 = 0

x
2
+ 9x - 3x - 27 = 0

=> x(x + 9) - 3(x - 9) = 0

=> (x - 3)(x + 9) = 0

=> either x - 3 = 0 or x + 9 = 0

=> x = 3 or x = -9

The values of x are (3, -9).
Question 2: Solve 8
x
= 32
Solution:
Given equation is 8
x
= 32.

=> 8
x
= 32

or (2
3
)
x
= 2
5


or 2
3x
= 2
5


Since the bases of the equation are same, set the exponents equal to one
another:

=> 3x = 5

x = 5353.
Question 3: Solve x+3−−−−−√x+3 = 2
Solution:
Given equation is x+3−−−−−√x+3 = 2

185. => x+3−−−−−√x+3 = 2

To isolate the x, squaring both side

=> x + 3 = 4

x = 4 - 3

x = 1.

How many types of equations are there in Algebra?

Students of algebra can become daunted when they imagine that there is
an infinite number of types of equations they must learn to solve in order to
master the solving of algebaic equations.
The question, as phrased, seems to assume that there is only one type of
algebra.

There are, in fact, many different types of algebra in mathematics,
depending on the specific mathematical discipline concerned, as can be
seen by the list of entries for 'algebra' at MathWorld (probably one of the
best resources for maths on the
web), http://mathworld.wolfram.com/sea.... The list of different types of
algebra goes on for some 8 pages, and is probably still not exhaustive.

The general definition of an algebra is a set of operations and
relations for combining and manipulating mathematical
entities (not just quantities), and the rules governing these.
(http://en.wikipedia.org/wiki/Alg...)

If by 'Algebra', the questioner means elementary, or high-school algebra,
then the mathematical entities concerned will usually be ordinary numbers,
i.e quantities, and the equations involving these can be categorized by such
criteria as the highest power of any variable in the equation:

186. Highest power - Name
 1 - linear equation
 2 - quadratic equation
 3 - cubic equation
 4 - quartic equation
 5 - quintic equation
 n - polynomial equation of degree n

and then further categorized by whether the coefficients in the equations
are integer,rational or irrational.

More advanced high-school mathematics might also include linear
algebra, which involves the solution of systems of coupled, simultaneous
linear equations, often involving the respective algebras
of matrices, determinants and possibly vectors, and the solution
of eigenvalue problems.

Then, of course, we can introduce complex numbers, and complex
variables, which also have their own algebra, and which can be
substituted for the real (i.e. non-complex) numbers and variables in any of
the above types of equations.

Finally, we can make the transition from polynomial equations of finite
degree to those of infinite degree - i.e. equations involving the sum of an
infinite number of terms. These are called infinite series, or
simply series, and introduce a whole new area of mathematics based
around the problem of finding the limiting value, or limit, of a series as
its dependent variable approaches a specific value.

As you can see, there are many different types of equation involved in high-
school algebra and the student must learn the specific techniques required
to solve each type (sorry, but there are no short cuts in a proper
mathematics education).

Once you move onto more advanced mathematics at college, you will learn
yet more types of equations and more types of algebras, some involving
ordinary numbers, but many involving highly abstract entities as far
removed from ordinary numbers as you can imagine.

187. But while the mathematics may become more abstract, it does not
necessarily follow that the actual mechanical manipulations required to
solve its equations will also become more complicated and time-consuming
(which is contrary to the popular misconception about advanced
mathematics). In fact, it is often the case that, as an area of mathematics
becomes more advanced, its notation becomes more abstract, powerful and
compact (you can say more with fewer symbols, basically), its equations
become shorter, not longer, and the rules for solving them actually
become simpler than those encountered in more elementary mathematics!
(It is similar to the difference between prose and poetry; you still use the
same words in poetry as you do in prose, but poetry can be more abstract
than prose, and often expresses far more than prose with far fewer words.)

So, in a sense, what makes advanced mathematics more difficult than
elementary mathematics is not so much the size and complexity of its
equations, and the amount of effort required to solve them, but simply the
level of conceptual abstraction that they represent. Your brain has to work
harder to understand them, but your writing hand doesn't have to work any
harder to solve the equations.

There are a few exceptions to this rule, however; in theoretical physics, the
equations frequently get longer and more complicated as the level of
conceptual abstraction increases, so not only do you have to think harder,
you also have to do considerably more work to solve the equations!
Happily, some of us are mathematical masochists; we actually enjoy the
challenge of grappling with, and the satisfaction of solving, horribly long,
complicated equations - often because the equation we get at the end is not
only powerful and profound, but also short, simple and very beautiful, as all
the most important equations in physics tend to be. It is this reduction of
messy mathematical complexity to elegant simplicity that is one of the
biggest buzzes you get doing theoretical physics. :o)

Solving Systems of Linear Equations
A system of linear equations is just a set of two or more linear equations.
In two variables (x and y), the graph of a system of two equations is a pair of lines
in the plane.

188. There are three possibilities:
 The lines intersect at zero points. (The lines are parallel.)
 The lines intersect at exactly one point. (Most cases.)
 The lines intersect at infinitely many points. (The two equations represent the
same line.)
Zero solutions:
y = –2x + 4
y = –2x – 3


One solution:
y = 0.5x + 2
y = –2x – 3


Infinitely many solutions:

189. y = –2x – 4
y + 4 = –2x

There are a few different methods of solving systems of linear equations:
1. The Graphing Method. This is useful when you just need a rough answer,
or you're pretty sure the intersection happens at integer coordinates. Just
graph the two lines, and see where they intersect!
See the second graph above. The solution is where the two lines intersect,
the point (–2, 1).
2. The Substitution Method. First, solve one linear equation for y in terms
of x. Then substitute that expression for y in the other linear equation.
You'll get an equation in x. Solve this, and you have the x-coordinate of the
intersection. Then plug in x to either equation to find the corresponding y-
coordinate. (If it's easier, you can start by solving an equation for x in terms
of y, also – same difference!)
Example:
Solve the system
Solve the second equation for y.
y = 19 – 7x
Substitute 19 – 7x for y in the first equation and solve for x.
3x + 2(19 – 7x) = 16

190. 3x + 38 – 14x = 16
–11x = –22
x = 2
Substitute 2 for x in y = 19 – 7x and solve for y.
y = 19 – 7(2)
y = 5
The solution is (2, 5).
3. The Linear Combination Method, aka The Addition Method, aka The
Elimination Method. Add (or subtract) a multiple of one equation to (or
from) the other equation, in such a way that either the x-terms or the y-
terms cancel out. Then solve for x (or y, whichever's left) and substitute
back to get the other coordinate.
Example:
Solve the system
Multiply the first equation by –2 and add the result to the second equation.
–8x – 6y = 4
8x – 2y = 12
–8y = 16
Solve for y.
y = –2
Substitute for y in either of the original equations and solve for x.
4x + 3(–2) = –2
4x – 6 = –2
4x = 4
x = 1

191. The solution is (1, –2).
4. The Matrix Method. This is really just the Linear Combination method,
made simpler by shorthand notation.
Properties of Inequalities
Inequality tells us about the relative size of two values.
(You might like to read a gentle Introduction to Inequalities first)
The 4 Inequalities
Symbol Words Example
> greater than x+3 > 2
< less than 7x < 28
≥ greater than or equal to 5 ≥ x-1
≤ less than or equal to 2y+1 ≤ 7


The symbol "points at" the smaller value

192. Properties
Inequalities have properties ... all with special names!
Here we list each one, with examples.
Note: the values a, b and c we use below are Real Numbers.

Transitive Property
When we link up inequalities in order, we can "jump over" the middle inequality.

If a < b and b < c, then a < c
Likewise:
If a > b and b > c, then a > c
Example:
 If Alex is older than Billy and
 Billy is older than Carol,
then Alex must be older than Carol also!

193. Reversal Property
We can swap a and b over, if we make sure the symbol still "points at" the
smaller value.
 If a > b then b < a
 If a < b then b > a
Example: Alex is older than Billy, so Billy is younger than Alex
Law of Trichotomy
The "Law of Trichotomy" says that only one of the following is true:

It makes sense, right? a must be either less than b or equal to b or greater
than b. It must be one of those, and only one of those.
Example: Alex Has More Money Than Billy
We could write it like this:
a > b
So we also know that:
 Alex does not have less money than Billy (not a<b)
 Alex does not have the same amount of money as Billy (not a=b)
(Of course!)

194. Addition and Subtraction
Adding c to both sides of an inequality just shifts everything along, and the
inequality stays the same.

If a < b, then a + c < b + c
Example: Alex has less coins than Billy.
If both Alex and Billy get 3 more coins each, Alex will still have less coins than
Billy.
Likewise:
 If a < b, then a − c < b − c
 If a > b, then a + c > b + c, and
 If a > b, then a − c > b − c
So adding (or subtracting) the same value to both a and b will not change the
inequality

Multiplication and Division
When we multiply both a and b by a positive number, the inequality stays
the same.
But when we multiply both a and b by a negative number, the
inequality swaps over!

195. Notice that a<b becomes b<a after multiplying by (-2)
But the inequality stays the same when multiplying by +3
Here are the rules:
 If a < b, and c is positive, then ac < bc
 If a < b, and c is negative, then ac > bc (inequality swaps over!)
A "positive" example:
Example: Alex's score of 3 is lower than Billy's score of 7.
a < b
If both Alex and Billy manage to double their scores (×2), Alex's score will still
be lower than Billy's score.
2a < 2b
But when multiplying by a negative the opposite happens:
But if the scores become minuses, then Alex loses 3 points and Billy loses
7 points
So Alex has now done better than Billy!
-a > -b
Why does multiplying by a negative reverse the sign?
Well, just look at the number line!

196. For example, from 3 to 7 is an increase, but from -3 to -7 is a decrease.

-7 < -3 7 > 3
See how the inequality sign reverses (from < to >) ?
Additive Inverse
As we just saw, putting minuses in front of a and b changes the direction of
the inequality. This is called the "Additive Inverse":
 If a < b then -a > -b
 If a > b then -a < -b
This is really the same as multiplying by (-1), and that is why it changes
direction.
Example: Alex has more money than Billy, and so Alex is ahead.
But a new law says "all your money is now a debt you must repay with hard
work"
So now Alex is worse off than Billy.

197. Multiplicative Inverse

Taking the reciprocal (1/value) of both a and
b can change the direction of the inequality.
When a and b are both positive or both
negative:
 If a < b then 1/a > 1/b
 If a > b then 1/a < 1/b

Example: Alex and Billy both complete a journey of 12 kilometers.
Alex runs at 6 km/h and Billy walks at 4 km/h.
Alex’s speed is greater than Billy’s speed
6 > 4
But Alex’s time is less than Billy’s time:
12/6 < 12/4
2 hours < 3 hours
But when either a or b is negative (not both) the direction stays the same:
 If a < b then 1/a < 1/b
 If a > b then 1/a > 1/b

198. Non-Negative Property of Squares
A square of a number is greater than or equal to zero:
a
2
≥ 0
Example:
 (3)
2
= 9
 (-3)
2
= 9
 (0)
2
= 0
Always greater than (or equal to) zero
Square Root Property
Taking a square root will not change the inequality (but only when both a and b
are greater than or equal to zero).
If a ≤ b then √a ≤ √b
(for a,b ≥ 0)
Example: a=4, b=9
 4 ≤ 9 so √4 ≤ √9

INEQUALITIES
←
Properties of Inequalities

199. →
page 1 of 2
Formal Definition of Inequalities
There are formal definitions of the inequality relations > , < ,≥,≤ in terms of the
familiar notion of equality. We say a is less than b , written a < b if and only if
there is a positive number c such that a + c= b . Recall that zero is not a positive
number, so this cannot hold if a= b . Similarly, we say a is greater than b and
write a > b if b is less than a ; alternately, there exists a positive number c such
that a = b +c .
The Trichotomy Property and the Transitive Properties of Inequality

Trichotomy Property: For any two real numbers a and b , exactly one of the
following is true: a < b , a = b , a > b .

Transitive Properties of Inequality:
If a < b and b < c , then a < c .
If a > b and b > c , then a > c .
Note: These properties also apply to "less than or equal to" and "greater than
or equal to":
If a≤b and b≤c , then a≤c .
If a≥b and b≥c , then ageqc .

Property of Squares of Real Numbers:
a
2
≥ 0 for all real numbers a .

200. Properties of Addition and Subtraction

Addition Properties of Inequality:
If a < b , then a + c < b + c
If a > b , then a + c > b + c
Subtraction Properties of Inequality:
If a < b , then a - c < b - c
If a > b , then a - c > b - c
These properties also apply to ≤ and ≥ :
If a≤b , then a + c≤b + c
If a≥b , then a + c≥b + c
If a≤b , then a - c≤b - c
If a≥b , then a - c≥b - c
Properties of Multiplication and Division
Before examining the multiplication and division properties of inequality, note the
following:
Inequality Properties of Opposites
If a > 0 , then - a < 0
If a < 0 , then - a > 0
For example, 4 > 0 and -4 < 0 . Similarly, -2 < 0 and 2 > 0 . Whenever we
multiply an inequality by -1 , the inequality sign flips. This is also true when
both numbers are non-zero: 4 > 2 and -4 < - 2 ; 6 < 7 and -6 > - 7 ; -2 <
5 and 2 > - 5 .
INEQUALITIES
←
Properties of Inequalities (page 2)

201. →
page 2 of 2
In fact, when we multiply or divide both sides of an inequality by any negative
number, the sign always flips. For instance, 4 > 2 , so 4(- 3) < 2(- 3) : -12 < - 6 . -
2 < 6 , so > : 1 > -3. This leads to the multiplication and division properties
of inequalities for negative numbers.

Multiplication and Division Properties of Inequalities for positive numbers:
If a < b and c > 0 , then ac < bc and <
If a > b and c > 0 , then ac > bc and >
Multiplication and Division Properties of Inequalities for negative numbers:
If a < b and c < 0 , then ac > bc and >
If a > b and c < 0 , then ac < bc and <
Note: All the above properties apply to ≤ and ≥ .
Properties of Reciprocals
Note the following properties:
If a > 0 , then > 0
If a < 0 , then < 0
When we take the reciprocal of both sides of an equation, something
interesting happens--if the numbers on both sides have the same sign, the
inequality sign flips. For example, 2 < 3, but > . Similarly, > , but -3
< . We can write this as a formal property:
If a > 0 and b > 0 , or a < 0 and b < 0 , and a < b , then >
If a > 0 and b > 0 , or a < 0 and b < 0 , and a > b , then <

202. Note: All the above properties apply to ≤ and ≥ .
Properties of inequality

We will show 6 properties of inequality. When appropriate, we will
illustrate with real life examples of properties of inequality.

Let x, y, and z represent real numbers

Addition property:

If x < y, then x + z < y + z

Example: Suppose Sylvia's weight < Jennifer's weight, then Sylvia's
weight + 4 < Jennifer's weight + 4

Or suppose 1 < 4, then 1 + 6 < 4 + 6

If x > y, then x + z > y + z

Example: Suppose Sylvia's weight > Jennifer's weight, then Sylvia's
weight + 9 > Jennifer's weight + 9

Or suppose 4 > 2, then 4 + 5 > 2 + 5

Subtraction property:

If x < y, then x − z < y − z

Example: Suppose Sylvia's weight < Jennifer's weight, then Sylvia's
weight − 4 < Jennifer's weight − 4

Or suppose 4 < 8, then 4 − 3 < 8 − 3

If x > y, then x − z > y − z

Example: Suppose Sylvia's weight > Jennifer's weight, then Sylvia's
weight − 9 > Jennifer's weight − 9

203. Or suppose 8 > 3, then 8 − 2 > 3 − 2

Multiplication property:

If x < y, and z > 0 then x × z < y × z

Example: Suppose 2 < 5, then 2 × 10 < 5 × 10 ( Notice that z = 10 and
10 > 0)

If x > y, and z > 0 then x × z > y × z

Example: Suppose 20 > 10, then 20 × 2 > 10 × 2

If x < y, and z < 0 then x × z > y × z

Example: Suppose 2 < 5, then 2 × -4 > 5 × -4 ( -8 > -20. z = -4 and -4 < 0
)

If x > y, and z < 0 then x × z < y × z

Example: Suppose 5 > 1, then 5 × -2 < 1 × -2 ( -10 < -2 )

Division property:

It works exactly the same way as multiplication

If x < y, and z > 0 then x ÷ z < y ÷ z

Example: Suppose 2 < 4, then 2 ÷ 2 < 4 ÷ 2

If x > y, and z > 0 then x ÷ z > y ÷ z

Example: Suppose 20 > 10, then 20 ÷ 5 > 10 ÷ 5

If x < y, and z < 0 then x ÷ z > y ÷ z

Example: Suppose 4 < 8, then 4 ÷ -2 > 8 ÷ -2 ( -2 > -4 )

If x > y, and z < 0 then x ÷ z < y ÷ z

204. Example: Suppose 5 > 1, then 5 ÷ -1 < 1 ÷ -1 ( -5 < -1 )

Transitive property:

If x > y and y > z, then x > z

Example: Suppose 10 > 5 and 5 > 2, then 10 > 2

x < y and y < z, then x < z

5 < 10 and 10 < 20, then 5 < 20

Comparison property:

If x = y + z and z > 0 then x > y

Example: 6 = 4 + 2, then 6 > 4

The properties of inequality are more complicated to understand than
the property of equality.

Allow yourself plenty of time as you go over this lesson.Any
questions about the properties of inequality, let me know.

Properties of Inequality
The following are the properties of inequality for real numbers. They are closely related to
the properties of equality, but there are important differences.
Note especially that when you multiply or divide both sides of an inequality by a negative
number, you must reverse the inequality.
PROPERTIES OF INEQUALITY
Anti reflexive Property
For all real numbers x,


205. Anti symmetry
Property
For all real numbers x and y,


Transitive Property
For all real numbers x, y, and z,
 if x < y and y < z, then x < z.
 if x > y and y > z, then x > z.
Addition Property
For all real numbers x, y, and z,
 if x < y, then x + z < y + z.
Subtraction Property
For all real numbers x, y, and z,
 if x < y, then x – z < y – z.
Multiplication
Property
For all real numbers x, y, and z,
 if x < y, then

 if x > y, then

Division Property
For all real numbers x, y, and z,
with z ≠ 0,
 if x < y, then

206.  if x > y, then


Solving Inequalities
Sometimes we need to solve Inequalities like these:
Symbol Words Example

> greater than x + 3 > 2
< less than 7x < 28
≥ greater than or equal to 5 ≥ x - 1
≤ less than or equal to 2y + 1 ≤ 7

Solving
Our aim is to have x (or whatever the variable is) on its own on the left of the
inequality sign:
Something like: x < 5
or: y ≥ 11
We call that "solved".

207. How to Solve
Solving inequalities is very like solving equations ... we do most of the same
things ...
... but we must also pay attention to the direction of the inequality.

Direction: Which way the arrow "points"
Some things we do will change the direction!
< would become >
> would become <
≤ would become ≥
≥ would become ≤
Safe Things To Do
These are things we can do without affecting the direction of the inequality:
 Add (or subtract) a number from both sides
 Multiply (or divide) both sides by a positive number
 Simplify a side
Example: 3x < 7+3
We can simplify 7+3 without affecting the inequality:
3x < 10

208. But these things will change the direction of the inequality ("<" becomes
">" for example):
 Multiply (or divide) both sides by a negative number
 Swapping left and right hand sides
Example: 2y+7 < 12
When we swap the left and right hand sides, we must also change the
direction of the inequality:
12 > 2y+7
Here are the details:
Adding or Subtracting a Value
We can often solve inequalities by adding (or subtracting) a number from both
sides (just as inIntroduction to Algebra), like this:
Solve: x + 3 < 7
If we subtract 3 from both sides, we get:
x + 3 - 3 < 7 - 3
x < 4
And that is our solution: x < 4
In other words, x can be any value less than 4.

209. What did we do?
We went from this:

To this:



x+3 < 7

x < 4

And that works well for adding and subtracting, because if we add (or
subtract) the same amount from both sides, it does not affect the inequality
Example: Alex has more coins than Billy. If both Alex and Billy get three more
coins each, Alex will still have more coins than Billy.
What If I Solve It, But "x" Is On The Right?
No matter, just swap sides, but reverse the sign so it still "points at" the
correct value!
Example: 12 < x + 5
If we subtract 5 from both sides, we get:
12 - 5 < x + 5 - 5
7 < x
That is a solution!
But it is normal to put "x" on the left hand side ...

210. ... so let us flip sides (and the inequality sign!):
x > 7
Do you see how the inequality sign still "points at" the smaller value (7) ?
And that is our solution: x > 7
Note: "x" can be on the right, but people usually like to see it on the left hand
side.
Multiplying or Dividing by a Value
Another thing we do is multiply or divide both sides by a value (just as
in Algebra - Multiplying).
But we need to be a bit more careful (as you will see).

Positive Values
Everything is fine if we want to multiply or divide by a positive number:
Solve: 3y < 15
If we divide both sides by 3 we get:
3y/3 < 15/3
y < 5
And that is our solution: y < 5

211. Negative Values

When we multiply or divide by a negative
number
we must reverse the inequality.

Why?
Well, just look at the number line!
For example, from 3 to 7 is an increase,
but from -3 to -7 is a decrease.

-7 < -3 7 > 3
See how the inequality sign reverses (from < to >) ?
Let us try an example:
Solve: -2y < -8
Let us divide both sides by -2 ... and reverse the inequality!
-2y < -8
-2y/-2 > -8/-2
y > 4
And that is the correct solution: y > 4

212. (Note that I reversed the inequality on the same line I divided by the negative
number.)
So, just remember:
When multiplying or dividing by a negative number, reverse the inequality
Multiplying or Dividing by Variables
Here is another (tricky!) example:
Solve: bx < 3b
It seems easy just to divide both sides by b, which would give us:
x < 3
... but wait ... if b is negative we need to reverse the inequality like this:
x > 3
But we don't know if b is positive or negative, so we can't answer this one !
To help you understand, imagine replacing b with 1 or -1 in that example:
 if b is 1, then the answer is simply x < 3
 but if b is -1, then we would be solving -x < -3, and the answer would be x
> 3
So:
Do not try dividing by a variable to solve an inequality (unless you know the
variable is always positive, or always negative).

213. A Bigger Example
Solve: (x-3)/2 < -5
First, let us clear out the "/2" by multiplying both sides by 2.
Because we are multiplying by a positive number, the inequalities will not
change.
(x-3)/2 ×2 < -5 ×2
(x-3) < -10
Now add 3 to both sides:
x-3 + 3 < -10 + 3
x < -7
And that is our solution: x < -7
Two Inequalities At Once!
How do we solve something with two inequalities at once?
Solve:
-2 < (6-2x)/3 < 4
First, let us clear out the "/3" by multiplying each part by 3:
Because we are multiplying by a positive number, the inequalities will not
change.
-6 < 6-2x < 12

214. Now subtract 6 from each part:
-12 < -2x < 6
Now multiply each part by -(1/2).
Because we are multiplying by a negative number, the inequalities change
direction.
6 > x > -3
And that is the solution!
But to be neat it is better to have the smaller number on the left, larger on the
right. So let us swap them over (and make sure the inequalities point
correctly):
-3 < x < 6
Basic Rules

In this section, you will learn how so solve inequalities. "Solving'' an inequality means
finding all of its solutions. A "solution'' of an inequality is a number which when
substituted for the variable makes the inequality a true statement.
Here is an example: Consider the inequality

When we substitute 8 for x, the inequality becomes 8-2 > 5. Thus, x=8 is a solution of
the inequality. On the other hand, substituting -2 for x yields the false statement (-2)-2
> 5. Thus x = -2 is NOT a solution of the inequality. Inequalities usually have many
solutions.
As in the case of solving equations, there are certain manipulations of the inequality
which do not change the solutions. Here is a list of "permissible'' manipulations:

215. Rule 1. Adding/subtracting the same number on both sides.
Example: The inequality x-2>5 has the same solutions as the inequality x > 7.
(The second inequality was obtained from the first one by adding 2 on both
sides.)
Rule 2. Switching sides and changing the orientation of the inequality sign.
Example: The inequality 5-x> 4 has the same solutions as the inequality 4 < 5 -
x. (We have switched sides and turned the ``>'' into a ``<'').
Last, but not least, the operation which is at the source of all the trouble with
inequalities:
Rule 3a. Multiplying/dividing by the same POSITIVE number on both sides.
Rule 3b. Multiplying/dividing by the same NEGATIVE number on both sides
AND changing the orientation of the inequality sign.
Examples: This sounds harmless enough. The inequality has the same
solutions as the inequality . (We divided by +2 on both sides).
The inequality -2x > 4 has the same solutions as the inequality x< -2. (We
divided by (-2) on both sides and switched ">'' to "<''.)
But Rule 3 prohibits fancier moves: The inequality DOES NOT have
the same solutions as the inequality x > 1. (We were planning on dividing both
sides by x, but we can't, because we do not know at this point whether x will be
positive or negative!) In fact, it is easy to check that x = -2 solves the first
inequality, but does not solve the second inequality.
Only ``easy'' inequalities are solved using these three rules; most inequalities are
solved by using different techniques.
Let's solve some inequalities:
Example 1:
Consider the inequality

216. The basic strategy for inequalities and equations is the same: isolate x on one side, and
put the "other stuff" on the other side. Following this strategy, let's move +5 to the
right side. We accomplish this by subtracting 5 on both sides (Rule 1) to obtain

after simplification we obtain

Once we divide by +2 on both sides (Rule 3a), we have succeeded in isolating x on the
left:

or simplified,

All real numbers less than 1 solve the inequality. We say that the "set of solutions'' of
the inequality consists of all real numbers less than 1. In interval notation, the set of
solutions is the interval .
Example 2:
Find all solutions of the inequality

Let's start by moving the ``5'' to the right side by subtracting 5 on both sides (Rule 1):

or simplified,

How do we get rid of the ``-'' sign in front of x? Just multiply by (-1) on both sides
(Rule 3b), changing " " to " " along the way:

217. or simplified

All real numbers greater than or equal to -1 satisfy the inequality. The set of solutions
of the inequality is the interval .
Example 3:
Solve the inequality

Let us simplify first:

There is more than one route to proceed; let's take this one: subtract 2x on both sides
(Rule 1).

and simplify:

Next, subtract 9 on both sides (Rule 1):

simplify to obtain

Then, divide by 4 (Rule 3a):

and simplify again:

218. It looks nicer, if we switch sides (Rule 2).

In interval notation, the set of solutions looks like this: .
Exercise 1:
Find all solutions of the inequality

Answer.
Exercise 2:
Solve the inequality

Answer.
Exercise 3:
Solve the inequality

Answer.
Exercise 4:
Find all solutions of the inequality

Solving Inequalities: An Overview (page 1 of 3)
Sections: Linear inequalities, Quadratic inequalities, Other inequalities

219. Solving linear inequalities is very similar to solving linear equations, except for one small but important
detail: you flip the inequality sign whenever you multiply or divide the inequality by a negative. The easiest
way to show this is with some examples:
1)


Graphically, the solution is:

The only difference between the linear
equation "x + 3 = 2" and this linear
inequality is that I have a "less than"
sign, instead of an "equals" sign. The
solution method is exactly the same:
subtract 3 from either side.
Note that the solution to a "less than, but
not equal to" inequality is graphed with a
parentheses (or else an open dot) at the
endpoint, indicating that the endpoint is
not included within the solution.
2)


Graphically, the solution is:

The only difference between the linear
equation "2 – x = 0" and this linear
inequality is the "greater than" sign in
place of an "equals" sign.
Note that "x" in the solution does not
"have" to be on the left. However, it is
often easier to picture what the solution
means with the variable on the left. Don't
be afraid to rearrange things to suit your
taste.
3)


Graphically, the solution is:

The only difference between the linear
equation "4x + 6 = 3x – 5" and this
inequality is the "less than or equal to"
sign in place of a plain "equals" sign.
The solution method is exactly the
same.
Note that the solution to a "less than or
equal to" inequality is graphed with a
square bracket (or else a closed dot) at
the endpoint, indicating that the endpoint
is included within the solution.
4)


The solution method here is to divide
both sides by a positive two.
Copyright © Elizabeth Stapel 1999-2011 All Rights
Reserved

220. Graphically, the solution is:

5)


Graphically, the solution is:

This is the special case noted above.
When I divided by the negative two, I
had to flip the inequality sign.
The rule for example 5 above often seems unreasonable to students the first time they see it. But think
about inequalities with numbers in there, instead of variables. You know that the number four is larger
than the number two: 4 > 2. Multiplying through this inequality by –1, we get –4 < –2, which the number
line shows is true:

If we hadn't flipped the inequality, we would have ended up with "–4 > –2", which clearly isn't true.
Solving Inequalities: An Overview (page 2 of 3)
Sections: Linear inequalities, Quadratic inequalities, Other inequalities

221. The previous inequalities are called "linear" inequalities because we are dealing with linear expressions
like "x – 2" ("x > 2" is just "x – 2 > 0", before you finished solving it). When we have an inequality with
"x
2
" as the highest-degree term, it is called a "quadratic inequality". The method of solution is more
complicated.
 Solve x
2
– 3x + 2 > 0
First, I have to find the x-intercepts of the associated quadratic, because the intercepts are
where y = x
2
– 3x + 2 is equal to zero. Graphically, an inequality like this is asking me to find
where the graph is above or below the x-axis. It is simplest to find where it actually crosses the x-
axis, so I'll start there.
Factoring, I get x
2
– 3x + 2 = (x – 2)
(x – 1) = 0, so x = 1 or x = 2. Then the graph crosses the x-axis at 1 and 2, and the number line
is divided into the intervals (negative infinity, 1), (1, 2), and (2, positive infinity). Between thex-
intercepts, the graph is either above the axis (and thus positive, or greater than zero), or else
below the axis (and thus negative,
or less than zero).
There are two different algebraic
ways of checking for this positivity
or negativity on the intervals. I'll
show both.
1) Test-point method. The
intervals between the x-intercepts
are (negative infinity, 1), (1, 2),
and (2, positive infinity). I will pick
a point (any point) inside each
interval. I will calculate the value
of y at that point. Whatever the
sign on that value is, that is the
sign for that entire interval.
For (negative infinity, 1), let's say I
choose x = 0; then y = 0 – 0 + 2 = 2, which is positive. This says that y is positive on the whole
interval of (negative infinity, 1), and this interval is thus part of the solution (since I'm looking for a
"greater than zero" solution).
For the interval (1, 2), I'll pick, say, x = 1.5; then y = (1.5)
2
– 3(1.5) + 2 = 2.25 – 4.5 + 2 =
4.25 – 4.5 = –0.25, which is negative. Then y is negative on this entire interval, and this interval
is then not part of the solution.
For the interval (2, positive infinity), I'll pick, say, x = 3; then y = (3)
2
– 3(3) + 2 = 9 – 9 + 2 = 2,
which is positive, and this interval is then part of the solution. Then the complete solution for the
inequality is x < 1 and x > 2. This solution is stated variously as:

ADVERTISEMENT

222. inequality notation

interval, or set, notation

number line with parentheses
(brackets are used
for closed intervals)

number line with open dots
(closed dots are used
for closed intervals)
The particular solution format you use will depend on your text, your teacher, and your taste.
Each format is equally valid. Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved
2) Factor method. Factoring, I get y = x
2
– 3x + 2 = (x – 2)(x – 1). Now I will consider each of
these factors separately.
The factor x – 1 is positive for x > 1; similarly, x – 2 is positive for x > 2. Thinking back to when I
first learned about negative numbers, I know that (plus)×(plus) = (plus), (minus)×(minus) = (plus),
and (minus)×(plus) = (minus). So, to compute the sign on y = x
2
– 3x + 2, I only really need to
know the signs on the factors. Then I can apply what I know about multiplying negatives.
First, I set up a grid,
showing the factors
and the number line.

Now I mark the
intervals where each
factor is positive.

Where the factors
aren't positive, they
must be negative.

Now I multiply up the
columns, to compute
the sign of y on each
interval.

Then the solution of x
2
– 3x + 2 > 0 are the two intervals with the "plus" signs:

223. (negative infinity, 1) and (2, positive infinity).
 Solve –2x
2
+ 5x + 12 < 0.
First I find the zeroes, which are the endpoints of the intervals: y = –2x
2
+ 5x + 12 =
(–2x – 3)(x – 4) = 0 for x = –
3
/2 and x = 4. So the endpoints of the intervals will be at –
3
/2 and4.
The intervals are between the endpoints, so the intervals are (negative infinity,
–3
/2], [
–3
/2, 4], and
[4, positive infinity). (Note that I use brackets for the endpoints in "or equal to" inequalities,
instead of parentheses, because the endpoints will be included in the final solution.)
To find the intervals where y is negative by the Test-Point Method, I just pick a point in each
interval. I can use points such as x = –2, x = 0, and x = 5.
To find the intervals where y is negative by the Factor Method, I just solve each factor: –2x – 3 is
positive for –2x – 3 > 0, –3 > 2x, –3/2 > x, or x < –
3
/2; and x – 4 is positive for x – 4 > 0,
x > 4. Then I fill out the grid:

Then the solution to this inequality is all x's in