Mathemathicss in modern world .pptx

Mathematics, often seen as a dry and abstract subject, is in fact an indispensable tool that permeates every aspect of our modern world.

From the intricate calculations that power our smartphones to the complex algorithms that drive artificial intelligence, mathematics is the invisible force that shapes our lives.  

In the realm of technology, mathematics is the foundation upon which modern marvels are built. The algorithms that enable search engines to understand our queries and deliver relevant results are rooted in mathematical principles. The sophisticated software that powers our computers, smartphones, and other devices relies heavily on mathematical models and computations. Even the simple act of sending a text message involves intricate mathematical calculations to ensure that the message reaches its intended recipient.  

Beyond technology, mathematics plays a crucial role in fields such as science, engineering, and finance. In science, mathematical models are used to describe and predict natural phenomena. For example, physicists use mathematical equations to understand the behavior of subatomic particles, while biologists employ mathematical models to study the dynamics of ecosystems. In engineering, mathematics is essential for designing structures, analyzing materials, and optimizing processes. From bridges and skyscrapers to airplanes and automobiles, the safety and efficiency of these structures depend on sound mathematical principles.  

In the financial world, mathematics is indispensable for managing risk, analyzing market trends, and developing investment strategies. Financial institutions rely on complex mathematical models to assess creditworthiness, price derivatives, and manage their portfolios. The global economy is underpinned by a vast network of mathematical calculations that facilitate trade, investment, and economic growth.  

Moreover, mathematics has a profound impact on our daily lives in ways that we often take for granted. The GPS systems that guide our navigation are based on mathematical calculations that triangulate our location using satellite signals. The traffic lights that regulate our commutes are synchronized using mathematical algorithms to optimize traffic flow. Even the humble barcode on a product is a testament to the power of mathematics, as it encodes information about the product's identity and price.  

In conclusion, mathematics is an essential tool that shapes our world in countless ways. From the intricate workings of technology to the complex systems that govern our society, mathematics is the invisible force that drives progress and innovation. As we continue to advance as a civilization, the importance of mathematics will only grow, ensuring its enduring relevance in the modern world.

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SLIDE 2 OF 37 ER sro ous Dern y

PATTERN

A pattern is a structure, form, or design that
is regular, consistent, or recurring.
—

Dan

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DIFFERENT KINDS OF
PATTERN

SET = often: dictabl , never
quite repeatable, and often contain fractals

- provides an inexhaustible
supply of nature’s patterns; usually found in the

water, stone, and even growth of trees

- pattern found in
locomotion

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DIFFERENT KINDS OF WA h
PATTERN

is conceivably
the most basic pattern | in nature. Our hearts and
lungs follow a regular repeated pattern of sounds MA
or movement whose timing is adapted to our
body's needs.
- A texture is a quality of a Dar

certain object that we sense through touch.

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PATTERNS FOUND IN
NATURE

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PATTERNS FOUND IN
NATURE

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PATTERNS FOUND IN AY x
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NATURE

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— or mirror por
_— captures symm en the left
of a pattern is the same as the right half LAN

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PATTERNS FOUND IN NC h

NATURE

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- looks the same after por
some rotation (of less than one full turn) {

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PATTERNS FOUND IN
NATURE

- acquire symmetries when units
are repeated and turn out having identical
figures

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SYMMETRIES IN
NATURE

- Our body exhibits bilateral
symmetry. It can be divided into two identical
halves.

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SYMMETRIES IN
NATURE

present in animal movements

- Symmetry of motion is

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SYMMETRIES IN
NATURE

- exhibits bilateral symmetry

- cluster of radially symmetrical disk florets BILL

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SYMMETRIES IN
NATURE

- exhibits radial symmetry

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SYMMETRIES IN
NATURE

- exhibits wallpaper symmetry

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SEQUENCE

It refers to an ordered list of numbers called
terms, that may have repeated values. The
arrangement of these terms is set by a definite ME
rule.

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SOME TYPES OF
SEQUENCES

- To determine if the
series of numbers follow an arithmetic sequence,
check the difference between two consecutive
terms

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SOME TYPES OF
SEQUENCES

- To determine if the
series of numbers follow a geometric sequence,
check the quotient/ratio between two
consecutive terms

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SOME TYPES OF | \
SEQUENCES

- In this sequence, the
reciprocal of the terms behaved in a manner like
arithmetic sequence.

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SOME TYPES OF | h
SEQUENCES
The sequence is

organized in a way a en can be obtained by
adding the two previous numbers.

Fibonacci Series

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THE FIBONACCI IN

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THE FIBONACCI IN | h
NATURE -

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THE FIBONACCI IN
NATURE

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THE NATURE OF | h
MATHEMATICS

From our ancestor's realization of measures,
they were able to notice and recognize some
rudiment hints about patterns. mi

As a result, we made use of mathematics as
a brilliant way to understand the nature
comprehending the structure of its underlying
patterns and regularities.

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THE NATURE OF | »
MATHEMATICS

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APPLICATION OF
MATHEMATICS

It is our important tool in the field of
sciences, humanities, literature, medicine, and
even in music and arts; it is in the rhythm of our |
daily activities, operational in our communities,
and a default system of our culture.

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APPLICATION OF
MATHEMATICS

It helps us cook delicious meals by exacting
our ability to measure and moderately control of
heat.

MHI

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APPLICATION OF
MATHEMATICS

It also helps us to shop wisely, read maps,
use the computer, remodel a home with
constrained budget with utmost economy.

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MHI

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Vignette is a small impressionistic scene, an
illustration, a descriptive passage, a short essay, a
fiction or nonfiction work focusing on one
particular moment; or giving an impression
about an idea, character, setting, mood, aspect,
or object.

Vignette is neither a plot nor a full narrative
description, but a carefully crafted
that might be part of some larger work, or a
complete description in itself.

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Think of scenes in the real world portraying
a. Patterns and regularities in the world
b. Usefulness or application of mathematics

Write a vignette for each scene meaning
one vignette for (a) and one for (b). Don’t forget
to put a title for each vignette.

Submissions of this requirement will be in
our Google Classroom. A copy of the rubric that

will be used to grade the vignette will also be
provided.

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[TIER

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AX Aron Kim Asprec is presenting

It's already past five in the afternoon and I'
alone by the sand dunes of the seaside, feeling the
and hearing the tidal waves of the sea. This is my
after a very tiring day from the restaurant. Watc
and pos for magnificent starfishes along thi
makes me feel happy and relaxed. Tiredness easily fa

ce starts to inhibit within me. Before | went b
ven, | used to meditate for 10 to 15 minutes on
white sand just to make sure that I'll end this day'p
holistically stable. Breathing in, breathing out,
relaxed, the wind touches my face, this is indeed my

31. AX Aron Kim Asprec is presenting

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alone by the sand dunes of the seaside, feeling the
and hearing the üäai waves of the sea. This is my
after a very tiring day trom the restaurant. Watc
and looking for magnificent starfishes along th
makes me feel hapey and relaxed. Tiredness easily

ce starts to inhibit within me. Before | went ba
ven, | used to meditate for 10 to 15 minutes on

iust to make sure that I'll end this day =

holistically stable. Breathing in, breathing out, eye
relaxed, the wind honehas =>) ace, this is indeed my testi
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PRACTICE EXERCISES

à, 2,3} and H = (1, b,c). Define relation T from G to H as follows:
For all (x, y) € Gx H, (x,y) € T means that x and y are elements OMITE English alphabet
List down the elements of T.

is relation T a function? Why or why not?

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PRACTICE EXERCISES ays

Determine whether the operation defined per each item is a binary operation or not.

z (set of all integers that is divisible by 3). Define * on G, for any a, b € G,a*b=a+

2. Let G € Z (set of all integers). Define * on H, for any a,b € H,a «b =*

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Determine whether the operation defined per each item is a binary operation or not.

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RK we arb. A+

tof all integers), Define * on H. for any a,b € H,a

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LIDE 3

PRACTICE EXERCISES aye

Determine whether the operation defined per each item is a binary operation or not.

+ of all integers that is divisible by 3). Define * on G, for any a,b € G,a*

me SAT Arb= A + b

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2 (set of all integers). Define * on H, for any a,b € H,a*b ==

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IDE 3

PRACTICE EXERCISES av

Determine whether the operation defined per each item is a binary operation or not.

1. Let G € 32 (set of all integers that is divisible by 3). Define * on G, for any a,b € G,a =

IN ae AY axb= Alb

= 150 are +1

set of all integers). Define * on H, for any a,b € H, a + b =

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PRACTICE EXERCISES axle

Determine whether the operation defined per each item is a binary operation or not.

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PRACTICE EXERCISES ays

Determine whether the operation defined per each item is a binary operation or not.

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$ ARI ER
2 Let je Z (set of all ne oe. on FG CS OM
= GB axlb= a
b g

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PRACTICE EXERCISES av

Determine whether the operation defined per each item is a binary operation or not.

IR € 37 (set of all i Fe DE sible by 3). Define * on G, for any a, b € G,a » b = a + b.

$ ar nik atle € IN

ll. A un
Define * on H, for any a,b € H,d

2. Let (pg Z (set of all intés
a a eee ee

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PRACTICE EXERCISES av

Determine whether the operation defined per each item is a binary operation or not.

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En We adh ot € 2 ~~

2. Let fe Z (set of all ei oe. oF * on H, for CS ed
Re |
bad lo

axb= a.

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PRACTICE EXERCISES Gy"

Determine whether the op: efined per each item is a binary operation or not

1. Let G € 3Z gers that is divisible by 3). Define * on G, for any a,b € G,c

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e. A un
égers), Define * on H, for any a,b

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PRACTICE EXERCISES

Determine whether the operation defined per each item is a binary operation or not.

ers that is divisible by 3). Define * on G, for ar Zaxb=athb

z (set of all integers). Define * on H, for any a,b € H, + b =~

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SLIDE 2 OF 16 Sik A Ink Tools > 2 Blank Screen

INDUCTIVE REASONI

The type of reasoning that forms a
conclusion based on the examination of specific
examples is called

It uses a set of specific a to
reach an overarching conclusion or it is the
process of recognizing or observing patterns and
drawing a conclusion.

So in short, is the
process of reaching a general conclusion by
examining specific examples.

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SLIDE 3 OF 16 Zum ink Tools + 2 Blank Screen — UE End Show

DEDUCTIVE REASON

is the process of
reaching specific conclusion by applying general
ideas or assumptions, procedure or principle or it
is a process of reasoning logically from given

statement to a conclusion. Sa

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TWO TYPES OF REASO

a. Jennifer always leaves for school at 7:00 a.m. Jennife
on time. Jennifer assumes, then, that if she leaves at
school today, she will be on time.

b. Red meat has iron in it, and beef is red meat. Th
has iron in it.

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POLYA'S FOUR STEPS
PROBLEM SOLVING

Understand the problem
Devise a plan
Carry out the plan

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SLIDE 8 OF 16 EM 4 Pons
POLYA'S FOUR STEPS
PROBLEM SOLVING /

The sum of three consecutive positive integers is 165. What 2
are these three num!

When we say consecutive numbers,
succeeding numbers. Say, 4, 5, 6 are three conse
for single-digit numbers. For the two digit numb
these three consecutive is 32, 33, and 34. Noticing
number added by 1 from the first number and the
increased by 2 from the first number.

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POLYA'S FOUR STEPS
PROBLEM SOLVING

The sum of three consecutive positive integers is 165. What are
these three numbers?

Since we do not know what are these three co!
integers, we will be using a variable, say x to repre
number. This variable x zould be the first number.
consecutive, the second number will be increa: y 1. $6;
representation would be x + 1. The third number was
from the first number so the possible representation wou

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POLYA'S FOUR STEPS
PROBLEM SOLVING

The sum of three consecutive itive integers is 165. What are
these three numbers? pa /

Since, based on the problem that the sum of these t
positive integers is 165, the working equation is:

(x) + (X + 1) + (x + 2) = 165
where x be the first positive integer, x + 1 be the second
+ 2 be the third number.

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POLYA'S FOUR STEPS
PROBLEM SOLVING

The sum of three consecutive positive integers is 165. What
are these three num

We already know the working formula. “To
determine the three positive consecutive integers, We
the concept of Algebra here in order to solve\th
Manipulating algebraically the given equation;

x+x+1+x+2= 165
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POLYA'S FOUR STEPS
PROBLEM SOLVING

The sum of three consecutive positive integers is 165. What /
are these three numbers?

Combining similar terms; (+ 3y Ñ \ °

3x + 3 = 165
Transposing 3 to the right side of the equation;
3x = 165-3 s

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YA'S FOUR STEPS
BLEM SOLVING

sum of three consecutive positive integers is 165, What
hree numbers?

= FES
g, 3x = 162

a
oth side by 3 to determine the value of x A=

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POLYA'S FOUR STEPS
PROBLEM SOLVING

The sum of three consecutive positive integers is 165. What

are these three numbers?
t > à, 1 =
h \

SAN

™

x = 54 would be the first number. Now, the second

and we already know the value of x = 54. So, the

55. Then the third number would be x + 2 and agai

x = 54 so the third number is 56. Hence, the

consecutive integers whose sum is 165 are 54, 55 and
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POLYA'S FOUR STEPS
PROBLEM SOLVING

The sum of three consecutive positive integers is 165. What
are these three numbers?

D
Adding these three numbers, 54 + 55 + 56 wi EN
f 165.

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RECREATIONAL PROBLEMS!”
USING MATHEMATICS / \\\ /,

With the use of pencil or pen, connect by means of a line /
the nine dots (see figure below) without lifting a pen and re- 4 N
tracing the line. f

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57. A Aron kim Asprec is presenting

Ena HH PRE

RECREATIONAL PROBL
USING MATHEMATICS /
With the use of pencil or pen, connect by means of a line /

the nine dots (see figure below) without lifting a pen and re- N
tracing the line.

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MODULAR

ARITHMETIC

Mathematics in Modern World

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SLIDE 1 OF 19

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MODULAR “ee?

ARITHMETIC

Mathematics in Modern World

1È

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SLIDE 2 OF 19

MODULAR ARITHMETIC
Two integers a and ar said to be congruent modulo n whergh EN}

a-b
n

ar =. =| D) Ba
ha

is an integer.

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MODULAR ARITHMETIC

Two integers a and b are said to be congruent modulo n where n € N if
a-b
n
is an integer.

In this case, we write a = b (mod n).
The number n is called the modulus.

The statement a = b (mod n) is called a congruence.

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MODULAR ARITHMETIC

Two integers a and b are said to be congruent modulo n where n € N if
a-b
n
is an integer.

In this case, we write a = b (mod n).

The number n is called the modulus.

The statement a = b (mod n) is called a congruence. ER

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SLIDE 3 OF 19

MODULAR ARITHMETIC

EXAMPLE;

b
ls 53= E =
ech Le pe
>.

A

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ogle.com/vgv-cmrw-ndo?authus

A Aron Kim Asprec is presenting

MODULAR ARITHMETIC

EXAMPLE:
Is 53 = 17(mod 3)?

One way of checking

12 is an integer.

Hence, 53 = 17(mod 3).

AG

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Untitled document - Google

65. FA GEd 102: Mathematicsinthel X La x | [E Untitled document - Google

C am

ogle.com/vgv-cmrw-ndo?authus

A Aron Kim Asprec is presenting

MODULAR ARITHMETIC

EXAMPLE:
Is -17 = 4(mod 3)?

One way of checking

—7 is an integer.

Hence, —17 = 4(mod 3).

ag

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Untitled document - Google

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A Aron kim Asprec is presenting

MODULAR ARITHMETIC

Two integers a and b are said to be congruent modulo n where n € N if
a-b

n
is an integer.

This can be stated in the form,

lfa,b € Z and n € Z*, then a = b (mod n) if and only if n | a — b.

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FA GEd 102: Mathematicsinthel X La

ogle.com/vgv-emrw-ndo?aut

C am

A Aron Kim Asprec is presenting

MODULAR ARITHMETIC

EXAMPLE:
Is —17 = 4(mod 3)?

Another way of checking
nla-b=3|(-17-4)=3|-21

Hence, —17 = 4(mod 3).

Aa

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Untitled document - Google

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MODULAR ARITHMETIC

Another way to be able to write in a congruence modulo n is by dividing by n
and take the remainder. Let us say c = 3. Then

CT

“14(mod 3) = 2 2 \ |
=

AG

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Untitled document - Google

x| Oe

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MODULAR ARITHMETIC

Another way to be able to write in a congruence modulo n is by dividing by n
and take the remainder. Let us say c = 3. Then

14(mod 3) = 2 | mod a — {
fc] . —
1(mod 3) = 1 =>

la

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ogle.com

A Aron kim Asprec is presenting

SLIDE 8 OF 19

MODULAR ARITHMETIC

Another way to be able to write in a congruence modulo n is/bf dividing by n
and take the remainder. Let us say c = 3. Then

14(mod 3) = 2 3 \ A

1(mod 3) =1 | | oS

-5(mod 3) = 1

AG

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ogle.com

A Aron kim Asprec is presenting

SLIDE 8 OF 19

MODULAR ARITHMETIC

Another way to be able to write in a congruence médulo n is by dividing by n
and take the remainder. Let us say c = 3. Then

14(mod 3) = 2 5 m od %

1(mod 3) = 1

-5(mod 3) = 1

AG

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A Aron kim Asprec is presenting

SU oF 19

MODULAR ARITHMETIC © : | A

Another way to be able to write in a congruence modulo n is by dividing by n
and take the remainder. Let us say ¢% 3. Then

—~5 mod?

1(mod 3) =1 er > b
————
-5(mod 3) = 1 TA

14(mod 3) = 2

210

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ogle.com

A Aron kim Asprec is presenting

SLIDE 8 OF 19

MODULAR ARITHMETIC © : | A

Another way to be able to write in a congruence modulo n is by dividing by n
and take the remainder. Let us say e*% 3. Then

nus ud >

1(mod 3) =1 a == b
Be
-5(mod 3) = 1 TA

la

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FA GEd 102: Mathematicsinthel X La Untitled document - Google D x

MODULAR ARITHMETIC © : | A

Another way to be able to write in a congruence modulo n is by dividing by n
and take the remainder. Let us say ¢% 3. Then

—~5 mod?

1(mod 3) = 1 a u L )
ee
-5(mod 3) = 1 AN

14(mod 3) = 2

AG

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Cc a

A Aron kim Asprec is presenting

MODULAR ARITHMETIC

PROPERTIES ON CONGRUENCE
Let n > 0 be fixed and a,b,c, d be arbitrary integers. Then
a.)a = a (mod n)
b.) if a = b (mod n) then b = a (mod n)
c.)ifa = b (mod n) and b = ¢ (mod n), then a = c (mod n)
d.)ifa = b (mod n) and c = d (mod n), then a + € = b + d (mod n) & ac = bd (mod n)
e.)if a = b (mod n), then a +c = b+ c (mod n) & ac = be (mod n)

f)ifa = b (mod n), then a* = b* (mod n)

AG

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76. FA Ged 102: Mathematics inthe) X | La Meet - vov © x | [El Untitled document- Google x | [Ej Untitled document - Google x | [E] cr

> (> & meetgoogle.com/vg
MR Aron kim Asprec is presenti
Aspı ing

SLIDE 10 OF 19

OPERATIONS ON i’
MODULAR ARITHMETIC .

ADDITION MODULO n
Evaluate (23 + 38) mod 12

A

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FA GEd 102: Mathematicsin the! X La Meet - vor

© à mestgoogle.com/va

A Aron kim Asprec is presenting

SLIDE 11 OF 19

OPERATIONS ON
MODULAR ARITHMETIC

SUBTRACTION MODULO n
Evaluate (12 — 7) mod 4

A

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A Aron kim Asprec is presenting

SLIDE 11 OF 19

OPERATIONS ON
MODULAR ARITHMETIC

SUBTRACTION MODULO n
Evaluate (12 — 7) mod 4

A

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A Aron kim Asprec is presenting

SLIDE 12 OF 19

OPERATIONS ON
MODULAR ARITHMETIC

SUBTRACTION MODULO n
Evaluate (14 — 21) mod 5. E) 1D A 5

Solution:

Subtracting 21 from 14, the result is —13.

On that case, we must find x so that —13 = x mod 5.

Trying the whole number values of x less than 5, the modulus, ie. x =0,1,23,4
-13-0 -13-1 -13-2 -13-3 -13-4
ge es s € qu 5
So, the only value for x is 2. Hence (14 — 21) mod 5 = 2.

EZ

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80. FS Ged 102: Mathematics inthe) X La Meet - v © x [El Untitled document- Google x | [Ej Untitled document - Google x | Cn

A Aron Kim Asprec is presenting

OPERATIONS ON
MODULAR ARITHMETIC

MULTIPLICATION MODULO n
Evaluate (15 - 23) mod 11.
Solution:

Multipying 15 : 23 to produce 345. To evaluate 345 mod 11, divide 345 by modulus,
11. The answer is the remainder.

A

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Cc a

A Aron kim Asprec is presenting

OPERATIONS ON
MODULAR ARITHMETIC

MULTIPLICATION MODULO n
Evaluate (15 - 23) mod 11

Solution:

Multipying 15 : 23 to produce 345. To evaluate 345 mod 11, divide 345 by modulus,
11. The answer is the remainder.

So (15 : 23) mod 11 = 4 since 345 = (11)(31) + 4 where 4 is the remainder.

AG

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ogle.com/vgv-emrw-ndo?authuser=0

A Aron Kim Asprec is presenting

SLIDE 13 OF 19

OPERATIONS ON
MODULAR ARITHMETIC

MULTIPLICATION MODULO n
Evaluate (15 - 23) mod 11.

Solution:

Multipying 15 : 23 to produce 345. To evaluate 345 mod 11, divide 345 by modulus,
11. The answer is the remainder.

So (15 : 23) mod 11 = 4 since 345 = (11)(31) + 4 where 4 is the remainder.

AG

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A Aron Kim Asprec is presenting

OPERATIONS ON
MODULAR ARITHMETIC

ADDITIVE INVERSES IN MODULAR ARITHMETIC
The sum of a number and its additive inverse equals the modulus.
EXAMPLE:
1. In mod 11 arithmetic, the additive inverse of 5 is 6 since 5 + 6 = 11.
2. In mod 11 arithmetic, the additive inverse of 4 is 7 since 4 + 7 = 11.
3. In mod 13 arithmetic, the additive inverse of 5 is 8 since 5 + 8 = 13.

4. In mod 13 arithmetic, the additive inverse of 4 is 9 since 4+ 9 = 13.

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A Aron kim Asprec is presenting

OPERATIONS ON
MODULAR ARITHMETIC

MULTIPLICATIVE INVERSES IN MODULAR ARITHMETIC
EXAMPLE:

2. In mod 7 arithmetic, find the multiplicative inverse of 2.
Solution:

To find the multiplicative inverse of 2, solve the equation 2x = 1 mod 7 by
trying different natural number values of x less than the modulus.

AG

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Cc a

A Aron kim Asprec is presenting

OPERATIONS ON
MODULAR ARITHMETIC

MULTIPLICATIVE INVERSES IN MODULAR ARITHMETIC
EXAMPLE:

2. In mod 7 arithmetic, find the multiplicative inverse of 2.
Solution

To find the multiplicative inverse of 2, solve the equation 2x = 1 mod 7 by

trying different natural number values of x less than the modulus.

Here x = 0,1,2,3,4,5,6.

CATHERINE HOSENA has left the meeting
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SOLVING CONGRUENCE
EQUATION

Solve 3x + 5 = 3 mod 4.

Solution:
If x = 0, 3(0) + 5 = 3 mod 4 is not a true congruence.
fx = 1, 3(1) + 5 = 3 mod 4 is not a true congruence.
lf x = 2, 3(2)+ 5 = 3 mod 4 is a true congruence.

If x = 3, 3(3) + 5 = 3 mod 4 is not a true congruence.

Aa

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A Aron Kim Asprec is presenting

SOLVING CONGRUENCE
EQUATION

Solve 3x + 4 = 2x + 8 (mod 3).
Solution
3x +4 = 2x + 8 (mod 3)
3x — 2x = (8 — 4)(mod 3)
x = 4 mod 3
x=1

Hence, the solution is x = 1

A

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E > CG“ @ meetgoogle.com/vgv-cmi do?authuser=0

A Aron kim Asprec is presenting

APPLICATIONS OF
MODULAR

ARITHMETIC

Mathematics in Modern World

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A Aron Kim Asprec is presenting

SOME APPLICATIONS

CLOCK

We can use modular arithmetic in determining time in the future or in the past.

EXAMPLE:

1. If itis 11 o'clock and you have to finish your math homework in 18 hours, what hour
will it be at that time?

Solution ue
Solving for 11+18, the answer nea isto evaluate(29à od12. Dividing 29
by the modulus 12, since our clock is a R%Rour clock, the remalndér is 5

Hence, the time that the homework could it be finished is 5 o'clock.

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C am

ogle.com/vgv-cmrw-ndo?authus

A Aron Kim Asprec is presenting

SOME APPLICATIONS

CLOCK

We can use modular arithmetic in determining time in the future or in the past.

EXAMPLE:

2. If the time now is 5 o'clock, what time is it 10 hours ago?

Solution

Solving for 5-10, the result is -5. Evaluating -5mod12, the result is 7

AG

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91. FA GEd 102: Mathematicsinthel X La x | [E Untitled document - Google D x Untitled document - Google’ x | [E] cr

C am

ogle.com/vgv-cmrw-ndo?authus

A Aron Kim Asprec is presenting

SOME APPLICATIONS

CLOCK

We can use modular arithmetic in determining time in the future or in the past.

EXAMPLE:

2. If the time now is 5 o'clock, what time is it 10 hours ago?

Solution
Solving for 5-10, the result is -5. Evaluating -5mod12, the result is 7
Hence, the time 10 hours ago is 7 o'clock

AG

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A Aron Kim Asprec is presenting

SOME APPLICATIONS

DAY OF THE WEEK

We can also use modular arithmetic in determining the day of the week by
assigning a number for each day of the week as shown below.

Monday = 0
Tuesday = 1
Wednesday = 2
Thursday = 3
Friday = 4
Saturday = 5
Sunday = 6

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SOME APPLICATIONS |Z mod +

DAY OF THE WEEK
EXAMPLE:

1. Let us say today is Wednesday. What would be the day 11 days after Wednesday?

Solution:

The equivalent number for Wednesday is 2. Adding 2 and 11, we get 13

Evaluate 13mod 7. (Our modulus here is 7 since there are 7 days in a week).

Dividing 13 by 7, the remainder is 6. The day that we assigned the number 6 is
Sunday.

Hence, it will be Sunday 11 days after Wednesday.

AG

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A Aron Kim Asprec is presenting

SOME APPLICATIONS

DAY OF THE WEEK
EXAMPLE
2. Letussay today is Friday. What day was it 13 days ago?

Solution
The equivalent number for Friday is 4. Subtracting 4 and 13, we ge:

Evaluate -9mod7.(Our modulus here is 7 since there are 7 days in a week), the
result is 5. The day that we assigned the number 5 is Saturday.

Hence, it was Saturday 13 days ago

A

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[E Untitled document - Google © x Untitled document - Google L x | [E] cr

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A Aron Kim Asprec is presenting

SLIDE 7

SOME APPLICATIONS VALID mx

INTERNATIONAL STANDARD BOOK NUMBER (ISBN)

One of the applications of modular arithmetic is on how to check or how to
determine whether the ISBN (International Standard Book Number) is valid or not. The

ISBN consists of 13 digits and this was created to help to ensure that orders for books
are filled accurately and that books are catalogued correctly.

The first digits of an ISBN are 978 (or 979), followed by 9 digits that are divided
into three groups of various lengths. These indicate the country or region, the publisher,
and the title of the book. The last digit (13th digit) is called a check digit.

978 - 971 - 23 - 9357-0
978 | Eu 5 ] | 9387
The
first | - | Country/region publisher] - | Title

three of the

digits book |

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A Aron Kim Asprec is presenting

SOME APPLICATIONS

INTERNATIONAL STANDARD BOOK NUMBER (ISBN)

If we label the first digit of an ISBN as d,, the second digit as dz and so on to the
13th digit as das ‚then the check digit is given by the modular formula as

dy3 = 10 — (d, + 3dz + dz + 3d4 + ds + 3d6 + dz + 3dg + do + 3410 + dy, + 3dy2)mod 10

The given ISBN is valid if the computed d,; and the given d,;/check digit are
equal. In the case that our computed d,3 = 10, then the given check digit should be 0.

AG

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97. FA GEd 102: Mathematicsin the! X | La

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Untitled document - Google D x | [Ej Untitled document - Google

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la

SON RERICATIONS I? ALT AU

INTERNATIONAL STANDARD BOOK NUMBER (ISBN) — =
EXAMPLE

The ISBN of Richard Aufmann’s book entitled “Mathematics in the Modern
World" published by Rex Bookstore in 2018 is 978-971 - 23 - 9357 - 0. Is ISBN valid?

Solution

dis = 10 — (d, + 3d, + dz + 3d, + ds + 3d, + dz + 3dg + dy + 3d,0 + dy, + 3dı2)mod 10
dia = 10 —(9 +3(7) + 8 + 3(9) +7 + 3(1) +2 +3(3) +9 + 3(3) + 5 + 3(7))mod 10
dis =10-(94+214+8+4+274+7+4+3+4+24+9+4+9+4+9+4+5+21)mod 10

dia = 10 -(130 mod 107)

dis =10-0=10

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Cc a

A Aron kim Asprec is presenting

SOME APPLICATIONS

INTERNATIONAL STANDARD BOOK NUMBER (ISBN)
EXAMPLE

The ISBN of Richard Aufmann's book entitled “Mathematics in the Modern
World" published by Rex Bookstore in 2018 is 978 - 971 - 23 - 9357 - 0. Is ISBN valid?

Solution:
dis = 10—0 = 10

Since our computed d,3 = 10 and our given check digit is O, hence our ISBN is valid.

210

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Cc a

A Aron kim Asprec is presenting

SOME APPLICATIONS

UNIVERSAL PRODUCT CODE (UPC)

Universal Product Code is placed on many items and is particularly useful in
grocery stores. The UPC is a 12-digit number that satisfies a modular equation that is
similar to the one for ISBNs. The last digit is the check digit.

The formula

di2 = 10 — (3d, + dz + 3d; +d, + 3ds + dé + 3d, + dg + 3d9 + dyo + 3d,,)mod 10

The given UPC is valid if the computed d and the given di2/check digit are
equal. In the case that our computed d,2 = 10, then the given check digit should be 0.

la

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Cc a

@ Aron Kim Asprec is presenting

SLIDE 13 OF 26

SOME APPLICATIONS

UNIVERSAL PRODUCT CODE (UPC)
EXAMPLE
The staple wire has a bar code of 9-02870-766290. Is the UPC number of this

product a valid number?

Solution

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101. FA Géd 102: Mathematics inthe Mo X La

GC à me:

gle.com,

A Aron Kim Asprec is presenting

GROUP THEORY

A group is a set of elements, with one operation and it must satisfy the following
four properties:

P1 : The set is closed with respect to the operation
For alla,b €G,thena*beG.

P2 : The operation satisfies the associative law.

For all a,b,c € G, then (a+b) +c =a~*(b«c).
P3 : There must be an identity element.
For everye € G,thene+a=a+eforallae G.
: Each element has an inverse.
For each a € G, then for every a? € G such that a » a Fe

=a lka=e

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102. FA Gta 102: Mathematics in the Mc X La

e

am

ogle.com/vgv-cmrw-ndo?authuse

A Aron kim Asprec is presenting

AG

GROUP THEORY

EXAMPLE
Show that neous sn addition as the operation form a group.

Solution:
P1 : Let a,b € Z. Now a + b € Zand (-a) + (—b) E Z. Hence, itis closed under addition

P2 : Let a,b,c € Z. The associative property of addition holds true for the integers.

P3 : The identity element of Zis 0 and 0 is an integer. Hence, there is an identity for
addition.

P4 : Each element of Z has an inverse that is if a € Z then -a is the inverse of a

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103. FA Géd 102: Mathematics inthe Mo X | La Meet -

> (o meet.google.com/vgv-cmi ‚do?authuse:
A Aron Kim Asprec is presenting

DE 4 OF

SYMMETRY OF GROUPS

Symmetry group is another type of group and it is based on regular polygon
(polygon whose sides are on the same length and with the same angle measure).

N
A

R120

Original Rotate 120 Rotate 240 Flip and Flip and
Position degrees degrees rotate rotate twice

2:41 PM | vgv-cmrw-ndo

104. FA Ged 102: Mathematics in the Mc X La Meet -

G & meetgoogle.com/vg
MR Aron kim Asprec is presenti
Aspı ing

DE 4 OF

GROUP THEORY Aeywven-
—

SYMMETRY OF GROUPS

Symmetry group is another type of group and it is based on regular polygon
(polygon whose sides are on the same length and with the same angle measure).

N I J IX
AAAAAA

A
R120 R240

Rf R2f
Original tate 120 Rotate 240 Flip and Flip and
Position degrees degrees rotate rotate twice

AG

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105. FA Ged 102: Mathematics inthe Mo X Le

C am

ogle.com/vgv-emrw-ndo?authuser=0

A Aron Kim Asprec is presenting

GROUP THEOR

SYMMETRY OF GROUPS

Symmetry group is another type of groufidad it/s based on regular polygon
(polygon whose sides are on the same length and With the same angle measure).

IN y À IN IN
/ S| à / vs
NA E N i \

R240 f Rf R2f

Original Rotate 120 Rotate 240 Flip Flip and Flip and
Position degrees degrees rotate rotate twice

\
\

WA

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106. FA Ged 102: Mathematics inthe Mo X Le

C am

A Aron Kim Asprec is presenting

GROUP THEOR

SYMMETRY OF GROUPS

ogle.com/vgv-emrw-ndo?authuse

ld

a
BNW

Symmetry group is another type of group and it is based on regular polygon
(polygon whose sides are on the same length and with the same angle measure).

A \ IN
\ Á : \ / | \
MM

N A A
NIN
> YN \
N N

1 R120
Original Rotate 120

Position degrees degrees

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Flip and Flip and
rotate rotate twice

107. FA Ged 102: Mathematics inthe Mo X Le

C am

ogle.com/vgv-emrw-ndo?authuser=0

A Aron Kim Asprec is presenting

GROUP THEORY

SYMMETRY OF GROUPS

Symmetry group is another type of group affgLitis based on regular polygon
(polygon whose sides are on the same length and with the same angle measure).

A A A EN A
N A \
A IN A À 2 h

AN

R120

Original Rotate 120 Rotate 240 Flip and Flip and
Position degrees degrees rotate rotate twice

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108. FA Ged 102: Mathematics inthe Mo X Le

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ogle.com/vgv-emrw-ndo?authuser=0

A Aron Kim Asprec is presenting

GROUP AN
SYMMETRY OF GROUPS E) A

Symmetry group is another type of group@r

orgregulaf polygon

(polygon whose sides are on the same length and with the same angke measure).

\

AL WA

JA IN
\ \
4 ww
A

A AN A À
IN 4 3\ /2 WN
4 \ h / \ y 4 N
À 2 À

| A.

I R120 R240 f

Original Rotate 120 Rotate 240 Flip
Position degrees degrees

2:51PM | vgv-emrw-ndo

Rf R2f

Flip and Flip and
rotate rotate twice

109. FA Ged 102: Mathematics inthe Mo X Le

C am

ogle.com/vgv-emrw-ndo?authuser=0

A Aron Kim Asprec is presenting

GROUP THEORY uN Te

Symmetry group is another type of group and fis based dE Er SL jon
(polygon whose sides are on the same length and with the same angle measure).

IN

R120 R240

Original Rotate 120 Rotate 240 Flip and Flip and
Position degrees degrees rotate rotate twice

2:53 PM | vgv-emrw-ndo

110. FA Géd 102: Mathematics inthe Mo X La

G à meet

ogle.com

A Aron kim Asprec is presenting

GROUP THEORY

A group must have an operation, a method by which two elements of the group
can be combined to form a third element that must also be a member of the group. The
operation we will use is called “followed by” and it symbolized by A

2:54 PM | vgv-emrw-ndo

111. GROUP THEORY
AN 3\ IN
AAA

4 /
A AL Ws

R120

EXAMPLE.

: rfl) a

a

2:56 PM | vgv-emrw-ndo

112. FA GEd 102: Mathematics in the Mo X La Meet - vgv

GC @i meet

ogle.com/vav-cmrw-ndo?authuse

A Aron kim Asprec is presenting

/\
1

h followed EY

EXAMPLE 4
1. Find FA R120. 4
Hence, FA R120 = Rf. R120

A

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113. A Aron Kim Asprec is presenting

SLIDE 7 OF 12

GROUP THEORY
1 3\ aN 1 À
N À \ y h " / A / h

2

I R120 R240 f Rf

EXAMPLE 972$ followed x
a

2. Find f A (RFA R120).

E

R120

A

3:02 PM | vg

114. FA GEd 102: Mathematics in the Mo XL

© à meta

le.com/vg

A Aron kim Asprec is presenting

EXAMPLE IL = À followed b;
2. Find f A (Rf A R120). 7 \ [m x

So, Rf A R120 = R2f. Rf

Then f A (RFA R120) = R240.

3:05 PM | vgv-emrw-ndo

115. FA Ged 102: Mathematics inthe Mo X Le

C am

ogle.com/vgv-emrw-ndo?authuser=0

A Aron Kim Asprec is presenting

ES
GROUP THEORY Gk lo = or

AN
\

EXAMPLE

2. Find FA (RF A R120).

So, RFA R120 = R2f. /
Then f A (RFA R120) = R240.

3:10 PM | vgv-emrw-ndo

116. FA Géd 102: Mathematics inthe Mo X La

G à meet

ogle.com

A Aron kim Asprec is presenting

GROUP THEORY

SYMBOLIC NOTATION

The operation notation for symmetry triangle can be presented into other
notation called as symbolic notation and/or permutation.

3:14 PM | vgv-cmrw-ndo

117. FA Géd 102: Mathematics inthe Mo X La

G à meet

ogle.com

A Aron kim Asprec is presenting

GROUP THEORY

SYMBOLIC NOTATION

The operation notation for symmetry triangle can be presented into other
notation called as symbolic notation and/or ee

LH, EE UE

3:16 PM | vgv-emrw-ndo

118. FA GEd 102: Mathematics in the Mc X | La Meet -

@ à mestgoogle.com/va

A Aron kim Asprec is presenting

GROUP THEORY
Éd oe sa

el 52) E)

EXAMPLE:

1.FindAAB. (Ls 3
(2

a

3:21PM | vgv-emrw-ndo

119. FA GEd 102: Mathematics in the Mc X | La Meet -

@ à mestgoogle.com/va

A Aron kim Asprec is presenting

GROUP THEORY

is AGD!
125 re

2
1
Di. 20:
nr D=( 3 oe

EXAMPLE:

st) CUY?)

a

3:22 PM | vgv-emrw-ndo

120. FA Ged 102: Mathematics in the Mo X La Meet - vgv

> @ à mestgoogle.com/va

A Aron kim Asprec is presenting

GROUP THEORY

(3 3) 231)
(23 2:

aa 213

EXAMPLE:

1.FindAAB.
(2% SG) |

a

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122. A Aronkim Asprec is presenting

GROUP THEORY
=( 3 3)

2
3 3 2
2 8 a 2
~ - à à 2

al,
EXAMPLE: 2 3
as

1.FindA AB. LT) NR: (2 )

a

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123. FA GEd 102: Mathematics in the Mo X | LA Meet - vov

Cai meet

gle.com/vg

A Aron kim Asprec is presenting

EXAMPLE:
1. Find A A B.

2. Find BAD.

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124. FA GEd 102: Mathematicsinthe Mo X | Cx Meet - vav-emrw-ndo ox E

GC à me:

A Aron kim Asprec is presenting

GROUP THEORY

AN A A
IN A A A
/ / \ / À \

3 AL \A 2

R120

Find (R2f A R120) AI
. Find (R240 A 1) A (FA Rf).
Find 1 A (R120 A R26)

la

3:29 PM | vgv-emrw-ndo

125. FA GEd 102: Mathematics in the Mc X | La Meet -

@ à mestgoogle.com/va

A Aron kim Asprec is presenting

GROUP THEORY

=( 2 3) 2:

(139

1. Find (I A C) AD.
2. Find EA (EAC).
Find (B À D A (CAA).

3:30 PM | vgv-emrw-ndo

126. A Aron Kim Asprec is presenting

It is a collection of methods for planning
experiments, obtaining data, and then analyzing,
interpreting and drawing conclusions based on the data. c

A CT AronkimAsprec

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127. A Aron Kim Asprec is presenting

SLIDE 3 OF 41 Am ÁS Ink Tools + Blank Screen End Show x

TWO TYPES OF STATIS

Descriptive statistics consist of organizing and
summarizing data. Descriptive statistics describe data
through numerical summaries, tables, and graphs.

Inferential statistics uses methods that take a result
from a sample, extend it to the population, and measure
the reliability of the result. y

A CT Aron kim Asprec

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128. A Aron Kim Asprec is presenting

SLE 3 or a WEN 0 Sous vou E

Descriptive statistics consist of organizing and
summarizing data. Descriptive statistics describe data
through numerical summaries, tables, and graphs.

Inferential statistics uses methods t take a result
from a iple, extend it to the ‚popuiaier: , and measure
the reliability of oa tant

A rae dh Ma Br

CASAS
CARA abs of a
A CT Aron kim Asprec es

2:52 PM | vgv-emrw-ndo © © 8 80 © : ee

129. A Aron Kim Asprec is presenting

For the following statements, decide whether it
belongs to the field of descriptive statistics or inferential
statistics.

1. A badminton player wants to know his average score
for the past 10 games.

—
Descriptive Statistics

2. A car manufacturer wishes to estimate the average J
lifetime of batteries by testing a sample of 50 batteries.

Inferential Statistics

A GT Aronkimasprec

Se @@ee0'°@

130. A Aron Kim Asprec is presenting

For the following statements, decide whether it
belongs to the field of descriptive statistics or inferential
statistics. c

3. Based on tests conducted at the Pearl Harbor
Laboratory using a random sample of rats, it was
concluded that rats are smart and are comparable to
humans when it comes to decision making. J

Inferential statist

A CT AronkimAsprec

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131. A Aron Kim Asprec is presenting

- words or codes that represent a class or category

- express a categorical attribute

Quantit

ative Variables

- number that represent an amount or a count

- numerical data, sizes are meaningful and answer
questions such as “how many” or “how much”

A CT AronkimAsprec

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132. A Aron Kim Asprec is presenting

SLIDE Woran EEE PA vus FE

Y - YVUMM L

- can be classified as Len E

a. =— Variables are data that can be counted

b. ontinuous Variables are data can assume all
values between any two specific values like 0.5, 1.2, etc.
and data that can be measured ee

a
aa mo

A CT AronkimAsprec

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133. A Aron Kim Asprec is presenting

SLIDE 13 OF 41 BR sro - Zus Benson |

vi
Nom I WE 11 .

de Catt. amtó="wo order

- data that consists of names, labels or categories only

CANE pice I
- cafes nes — somt order
- data that arranged in some order, but differences
between data.

A CT AronkimAsprec

Su 00:00:08

134. A Aron Kim Asprec is presenting

SLIDE 14 OF 41 BY zur Somme Dés y

nterval Level ~ PATA x
Se ee tae IE:

Dd
- has no absolute zero Glas Pe ot Le A

Tarerdme. OC © 32

à je

A GT Aronkimasprec

nen @@ee0'°@

135. A Aron Kim Asprec is presenting

SLIDE 14 OF 41 Zu ink Too's + 12 Blank Screen TE End Show

interval Level

- has no absolute zero

Ratio Lever — IMC ced Hate
- has an absolute zero OÖ gr) AL

cat Lib
Aye
A CT Aron kim Asprec

een @@ee0'°@

136. A Aron Kim Asprec is presenting

Identify the level of measurement of variable for

each item. y

es IMA
4.10 CAC a6 newt 4 E
-Interval level 5 N Gr Ana

5. Baking temperature N
¡nin |

Interval variable

namencal A

SANS

A CT Aron kim Asprec

ren 00=.0:8

137. A Aron Kim Asprec is presenting

The measures of central tendency is to describe the
centrality of the distribution into a single numerical unit.
This single numerical unit must provide clear description L
about the common trait being observed in the
distribution of scores.

A CT Aronkimasprec

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138. A Aron Kim Asprec is presenting

- the arithmetic average of all the scores

- can be determined by adding all the scores together and
then by dividing by the total number of scores

- appropriate to use when the scores in the distribution
are well-balanced (there are no extreme high or no
extreme low scores / outliers)

- sensitive to outliers

A CT AronkimAsprec

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139. A Aron Kim Asprec is presenting

Compute the mean of the first exam score
of 10 students enrolled in Introductory
Statistics.

82 + 77 + 90 + 71 + 62 + 68 + 74 + 84 + 94 + 88
j=
10

A CT Aronkimasprec

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140. A Aron Kim Asprec is presenting

- the point that separates the upper half from the lower
half of the distribution

-middle point or midpoint of any distribution
- not affected by outliers

A CT Aronkimasprec

mn 00:00:08

141. A Aron Kim Asprec is presenting

What is the median of the first exam
score of 5 students enrolled in Introductory
Statistics?

In solving for the median, we have to
arrange the scores first in either ascending or
descending order.

A CT Aron kim Asprec

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142. A Aron Kim Asprec is presenting

What is the median of the first exam
score of 5 students enrolled in Introductory
Statistics?

If the number of scores in the distribution
n+1

is ODD, then the median is the —-th data,
where n is the number of scores.

A CT Aronkimasprec

3:45 PM | vav-cmrw-ndo ES ® [=] ÿ B

143. A Aron Kim Asprec is presenting

What is the median of the first exam
score of 5 students enrolled in Introductory
Statistics?

In this example, there are 5 score (which
is odd).

5+1

Hence the median is the “= = = © = 3rd
score which is 77. di

A CT Aron kim Asprec

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144. A Aron Kim Asprec is presenting

What is the median of the first exam
score of 6 students enrolled in Introductory
Statistics.

If the number of scores in the distribution
is EVEN, then the median is the average of the
two middle most scores.

A CT AronkimAsprec

3:47 PM | vgv-emrw-ndo ES ® [=] ÿ B

145. A Aron Kim Asprec is presenting

What is the median of the first exam
score of 6 students enrolled in Introductory
Statistics.

In this example, there are 6 scores (even).

Hence, the median is Beben = Be = 74,

A CT AronkimAsprec

3:48 PM | vgv-emrw-ndo ES ® [=] ÿ B

146. A Aron Kim Asprec is presenting

The following data represent the number of O-ring
failures on the shuttle Columbia for its 17 flights prior to
its fatal flight: "

0, 0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3

Solution: r

The mode is 0 because it occurs most frequently (11
times).

A CT AronkimAsprec

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147. A Aron Kim Asprec is presenting

SLIDE 29 OF 41 m Æ Ink Tools = 1 Blank Screen Enc Show x

- Interval Data can entertain the calculations of all three
measures of central tendency.

- For ordinal data. median can be used.

- For the nominal data, however, neither the mean nor
the median can be used. Nominal data are restricted by
simply using a number as a label for a category and the
only measure ntral tendency permissible for nominal
data is the mode.

A CT AronkimAsprec

SE eye 00:10:68

148. A Aron Kim Asprec is presenting

- If the interval data distribution is fairly well balanced, it
is appropriate to use the mean to measure the central
tendency.

- If the distribution of the interval data is skewed
(presence of outlier), you may either remove the outlier
or adopt the median.

- If the interval data distribution manifests a significant
clustering of scores, then consider to visually analyze the
scores and find the presence of dominant constant which
is the mode.

A CT AronkimAsprec

SE eye 00:10:68

149. A Aron Kim Asprec is presenting

The RANGE describes variability of scores by merely
providing the width of the entire distribution. The range
can be found by simply determining the difference "
between the highest score and the lowest score.

The range is not stable enough to indicate variability
because it is easily affected by outliers.

A CT AronkimAsprec

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150. A Aron Kim Asprec is presenting

In this example, the highest score is 94
and the lowest score is 62.

Hence, the range is
94 — 62 = 32

A CT AronkimAsprec

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151. A Aron Kim Asprec is presenting

It provides measurement about how much all of the
scores in the distribution normally differ from the mean
of the distribution. Unlike the range, which utilizes only
two extreme scores, SD employs every score in the
distribution.

A CT AronkimAsprec

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152. A Aron Kim Asprec is presenting

x
|
The formula for SD is computed as follows:
Ip x2
SZ pe — X2 M
\
where » X is the sum of the raw scores, N is the number
of scores in a distribution and X is the mean of the
distribution. J

A CT AronkimAsprec

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153. A Aron Kim Asprec is presenting

A distribution with small standard deviation shows
that the trait being measured is homogenous.

While a distribution with a large standard deviation
is indicative that the trait being measured is
heterogeneous.

A distribution with zero standard deviation implies
that scores are all the same.

A CT Aronkimasprec

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154. A Aron Kim Asprec is presenting

Variance is the square of the standard deviation.

If both standard deviation and variance manifest
large values then it means heterogenous distribution and
when they both manifest small values, they provide
similar outcomes about the homogeneity of the
distribution.

A CT AronkimAsprec

en 00:00:08

155. A Aron Kim Asprec is presenting

For the following statements, decide whether it
belongs to the field of descriptive statistics or inferential
statistics.

1. Based on a random sample of instructors in BeeEsYu,
it was found that the average amount spent on travel in
the previous year of the sampled instructors is 15,000
pesos,

2. From 2011-2018 graduation data, BeeEsYu had one
summa cum laude in 2013, 2014, 2017, and 2018, two in
2012, and none in 2011, 2015, and 2016.

A CT AronkimAsprec

eg 00:10:68

156. FA Géd 102: MathematicsintheMo X | LA Meet -

€ > C À mee

oogle.com/vgv-cm

A Aron Kim Asprec is presenting

SIMPLE AND COMPOUND STATEMENTS

A statement is a declarative sentence that is either true or false, but not both
true and false

Determine whether the following sentences are statements or not a statement.
|. Batangas is a province of the Philippines.
The first sentence is a true sentence. This is a statement.
2, People with ages from 21 to 59 years old cannot be infected by COVID-19 virus
The second sentence is a false sentence. This a statement.
3.x+1=5

The third sentence is either true or false depending on the value of x. This
sentence cannot be both true and false at the same time in a specific value of x
Therefore, this is a statement.

AG ae

2:40 PM | vgv-emrw-ndo

157. FA Géd 102; Mathematics inthe Mo X | LA Meet -

+ > ©

A meet.google.com/vav-cm

A Aron kim Asprec is presenting

E 2

SIMPLE AND COMPOUND STATEMENTS

A statement is a declarative sentence that is either true or false, but not both
true and false

Determine whether the following sentences are statements or not a statement.
|. Batangas is a province of the Philippines.

The first sentence is a true sentence. This is a statement.

2. People with ages from 21 to 59 years old cannot be infected by COVID-19 virus

The second sentence is a false sentence. This a statement.
+1=5, A y x rs
The third sentence is either true or false depending on the value of x.This

sentence cannot be both true and false at the same time in a specific value of x
Therefore, this is a statement.

2:40 PM | vgv-emrw-ndo

158. FA Géd 102; Mathematics inthe Mo X | LA Meet -

e

A meet.google.com/vav-cm

A Aron Kim Asprec is presenting

SIMPLE AND COMPOUND STATEMENTS

A statement is a declarative sentence that is either true or false, but not both
true and false

Determine whether the following sentences are statements or not a statement.
4. Open the door.
This is not a declarative sentence. Hence, this is not a statement

5.What is your mother’s name?

This is not a declarative sentence. This is not a statement.
6.This a false statement.

This sentence is not a statement because if we assume it to be a true sentence,
then it is false, and if we assume it to be a false sentence, then it is true. Statements
cannot be true and false at the same time.

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159. FA Géd 102; Mathematics inthe Mo X | LA Meet -

e

A meet.google.com/vav-cm

A Aron Kim Asprec is presenting

SIMPLE AND COMPOUND STATEMENTS

A simple statement is a statement that conveys a single idea.
A compound statement is a statement that conveys two or more ideas.

By connecting simple statements with words and phrases such as and, or, if
then, and if and only if creates a compound statement. These words and phrases are
called connectives.

EXAMPLE:

SIMPLE | will play Mobile Legend. I will go to school.

2:48 PM | vgv-cmrw-ndo

160. FA Géd 102; Mathematics inthe Mo X | LA Meet -

€ > C À mee

oogle.com/vgv-cm

A Aron Kim Asprec is presenting

SIMPLE AND COMPOUND STATEMENTS

A simple statement is a statement that conveys a single idea.
A compound statement is a statement that conveys two or more ideas.

By connecting simple statements with words and phrases such as and, or, if...
then, and if and only if creates a compound statement. These words and phrases are
called connectives.

NOTATIONS:
and
or
if then

if and only if

AG AB

2:48 PM | vgv-cmrw-ndo

161. FA Géd 102; Mathematics inthe Mo X | LA Meet -

€ > © à meet.google.com/vgv-cmrw-ndo?authuser=0

A Aron Kim Asprec is presenting

DE 6 OF 3

SIMPLE AND COMPOUND STATEMENTS

Let p be a statement.

The negation of p, denoted by ~p ‚(read as “not p“) is a statement that is false
if p is true and it is true if p is false

2;

AG ae

2:49 PM | vgv-cmrw-ndo

162. ox HE — +

FA Gta 102: Mathematics inthe Mo X | La

ogle.com/vgv-emrw-ndo?authuser=0

A Aron Kim Asprec is presenting

SIMPLE AND COMPOUND STATEMENTS

Let p be a statement.

The negation of p, denoted by ~p ‚(read as “not p“) is a statement that is false
if p is true and it is true if p is false

EXAMPLE:

Rachel is a Blue Badge umpire in Rachel is not a Blue Badge umpire in
table tennis. table tennis.

The color of the roof is not green. The color of the roof is green.

AG ae

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163. FA Géd 102; Mathematics inthe Mo X | LA Meet -

2. 0 mes

pogle.com/vgv-cmrw-ndo?au

A Aron kim Asprec is presenting

SIMPLE AND COMPOUND STATEMENTS

We can use letters to denote our statements.
I will play table tennis.
I will go to school

| will not do my assignment

Translate the following compound statements in symbolic form.
1.1 will play table tennis or | will go to school.
pVq

2.1 will go to school and | will not play table tennis

2:51PM | vgv-emrw-ndo

164. user=0 m

ITIFIERS AND THEIR NEGATION J

JOHN MARK PA...
NTIAL QUANTIFIERS

ese are used as prefixes to assert the existence of something. These include
some, and the phrases there exists and at least one @

DONNA HORNIL. NATHANIEL

13
ffee shops are open “
xists an integer n such that 3n > 120.

CHRISTINE JOY —

165. FA GEd 102: Mathematics inthe Mo X La Meet

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A Aron Kim Asprec is presenting

QUANTIFIERS AND THEIR NEGATION

UNIVERSAL QUANTIFIERS

These are used as prefixes to assert that every element of a given set satisfies
some conditions or to deny the existence of something. These include the words all,
_— 00

every, none, and no

—— —

EXAMPLE
|.All players are nice people.

2. No even integers are divisible by 3.

AG ae

2:55 PM | vgv-emrw-ndo

166. £ > CG à meet.google.com/vgv-cmrw-ndo?authuser=0

A Aron Kim Asprec is presenting

QUANTIFIERS AND THEIR NEGATION

NEGATION OF STATEMENTS INVOLVING QUANTIFIERS
|. Some coffee shops are open
Negation :All coffee shops are not open.

No coffee shops are open.

2.All players are nice people.

AG ae

2:59 PM | vgv-emrw-ndo

167. AUTOLOG

a GI Aronkmaspree

2:59 PM | vgv-emrw-ndo NW) (9 © Ÿ © : ©

168. ox y ne Ear.

FA Gta 102: Mathematicsinthe Mo X La

ogle.com/vgv-emrw-ndo?authuser=0

A Aron Kim Asprec is presenting

TRUTH VALUE AND TRUTH TABLE OF A
STATEMENT

TRUTH VALUE

The truth a simple statement is either true (T) or false (F). The truth

value of a depends on the truth values of its simple statements
and its connective

TRUTH TABLE

A truth table is a table that shows the truth value of a compound statement for
all possible truth values of its simple statements.

AG ae

3:01PM | vgv-cmrw-ndo

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A Aron Kim Asprec is presenting

TRUTH VALUE AND TRUTH TABLE OF A
STATEMENT a V

TRUTH TABLE OF THE DISJUNCTION

ir E
HE T
F T
F F

170. FA GEd 102: MathematicsintheMc X La Mes

= > C A meet.google.com/vgv-cmi

TRUTH WALUE AND TRUTH TABLE OF A

STATEMENT A > en
2 un ; ire es me a au 6 A v) = te E vr)
LTITIE E ar E
ie) A T y
AREAS E + | TE
Ele lee E
Sees
u =
AE a
= mn: a

Z
El
6
al

171. FA Gta 102: Mathematicsin the Mo X La

ogle.com/vg

A Aron Kim Asprec is presenting

TRUTH VALUE AND TRUTH TABLE OF A
STATEMENT

2. Construct the truth table for the statement (=p Ar) > (qVr).

AG ae

3:37 PM | vgv-emrw-ndo

172. A Aron Kim Asprec is presenting

TRUTH WALUE AND TRUTH TABLE OF A
STATEMENT

TRY THIS

Construct the truth table of the following statements.

lr e -(p => -~r)

2. [(~qAr) A (p > q)] > (~pVr)

173. FA Gta 102: Mathematics in the Mc X | La Meet @ x E. y

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LOGICALLY EQUIVALENT STATEMENTS

EQUIVALENT STATEMENTS
Let p and q are two statements.

Then p and q are said to be logically equivalent (or simply equivalent),
denoted by p =
of their simple statements

q, if they both have the same truth values for all possible truth values

EXAMPLE:

Show that the statements ~(p/q) and ~p V ~q are equivalent statements.

AG ae

3:38 PM | vgv-emrw-ndo

174. FA Gta 102: Mathematicsin the Mo X La

ogle.com/vg

LOGICALLY EQUIVALENT STATEMENTS

AG ae

3:38 PM | vgv-emrw-ndo

175. FA GE 102: Mathematics in the Mo X | La Meet - @ x E. y

€ > C À me

e.com/vgy-cmrw-ndo?at

A Aron Kim Asprec is presenting

LOGICALLY EQUIVALENT STATEMENTS

DE MORGAN’S LAW
For any statements p and q, the following hold.
a.~(pAq) = ~pV~q

b.~(pVr) = ~pA~r

De Morgan's Law states that the negation of the conjunction statement is
equivalent to the disjunction of the negation of each simple statement. And, the
negation of the disjunctive statement is equivalent to the conjunction of the negation of

each simple statement.

3:39 PM | vgv-emrw-ndo

176. FA Gi 102: Mathematics inthe Mo X | LA Meet -

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A Aron Kim Asprec is presenting

LOGICALLY EQUIVALENT STATEMENTS

The equivalent forms of conditional and biconditional statements are given
a. Conditional :p=>q=-pVq

b. Biconditional 'poqg=(p>g)Ag>p)

AG ae

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177. FA GE 102: Mathematics in the Mo X | La Meet - @ x E. y

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A Aron Kim Asprec is presenting

TAUTOLOGY AND CONTRADICTIONS

A tautology is a statement whose truth value is always true regardless of the
truth values of its individual simple statements. A statement which is a tautology is
called tautological statement.

A contradiction is a statement whose truth value is always false regardless of

the truth values of the individual simple statements that is a contradiction is a
contradictory statement.

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TRUTH VALUE AND TRUTH TABLE OF A
STATEMENT

Example of a tautological statement: (=p Ar) > (qVr).

Mal

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A Aron Kim Asprec is presenting

TRUTH VALUE AND TRUTH TABLE OF A
STATEMENT

Example of a contradictory statement: (pVr) A (=p A =r)
pl if ie P

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A Aron Kim Asprec is presenting
380 PH Te, May 17 m

€ Reñection.paf

REFLECTION VIDEO

Record a 1 to 3 minutes video of yourself answering the question,

How does Mathematics in the Modern World (MMW) change your
perception of Mathematics and its application in our everyday life?

Submit your videos in our google classroom.

Tum on captions (c)

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MATHEMATICS OF GRAPHS

Graph theory is an area of mathematics that focuses in the study of
structured graphs used to model pair wise relations between objects.

The concept of graph theory was first studied by the famous mathematician
Leonard Euler in 1735 by giving the solution to the famous problem known as the

Seven Bridges of Konisberg.

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A Aron Kim Asprec is presenting

MATHEMATICS OF GRAPHS

Graph theory is an area of mathematics that focuses in the study of
structured graphs used to model pair wise relations between objects.

The concept of graph theory was first studied by the famous mathematician
Leonard Euler in 1735 by giving the solution to the famous problem known as the
Seven Bridges of Konisberg.

Bridges of Königsberg

Can you take a walk through the town,
visiting each part of the town and crossing
each bridge only once?

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GRAPHS AND GRAPHS MODELS

GRAPH

A graph G is composed of two finite sets: a nonempty set V of vertices and a
set E of edges, where each edge is associated with a set consisting of either one or
two vertices called its endpoints.

EXAMPLE:
Let G be a graph with V = (a, b,c, d) and E = {[a, b], [b,d], [a, a], [b,c], [c, d]).

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A Aron Kim Asprec is presenting

GRAPHS AND GRAPHS MODELS

GRAPH

A graph G is composed of two finite sets: a nonempty set V ofvertices and a
set E of edges, where each edge is associated with a set consisting of either one or
two vertices called its endpoints.

EXAMPLE:
Let G be a graph with V = {a,b,c,d} and E = {[a, b], [b, d], (a, d], [b,c], [c, d]}.

This graph consists of 4 vertices and 5 edges.

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A Aron Kim Asprec is presenting

GRAPHS AND GRAPHS MODELS

EXAMPLE:
Let G a graph withV = {a,b,c,d} and E = {[a, b], [b, d], [a, d], [b, c], [c, d]}.
This graph consists of 4 vertices and 5 edges.

Non-empty set V is called the Vertex Set and set E is called the Edge set.

Vertices are sometimes called points or nodes and edges are sometimes
called lines.

In this example, we can also use the notation V(G) and E(G) to denote that
set V is the vertex set of graph G and E(G) is the edge set of graph G respectively.

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GRAPHS AND GRAPHS MODELS

GRAPH

If x and y are vertices of a graph G, we adopt the notation [x, y] to denote
an edge of G with endpoints x and y.

If [x, y] is an edge of a graph, x is adjacent to y or y is adjacent to x.

An edge [x, y] is said to be incident on the vertices x and y.

The number of elements of V is called the order of a graph and the number
of elements of E is called the size of a graph.

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A Aron Kim Asprec is presenting

GRAPHS AND GRAPHS MODELS

PICTORIAL REPRESENTATION OF A GRAPH
Let G be a graph where V(G)= {a,b,c,d} and E(G) = {[a, b], [b, d], [a, d], [b, c], [c, d]}.

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A Aron Kim Asprec is presenting

GRAPHS AND GRAPHS MODELS

PICTORIAL REPRESENTATION OF A GRAPH
Let G be a graph where V(G)= {a,b,c,d} and E(G) = {[a, b], [b, d], [a, d], [b, c], [c, d]}.

a @ @b

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A Aron Kim Asprec is presenting

GRAPHS AND GRAPHS MODELS

PICTORIAL REPRESENTATION OF A GRAPH
Let G be a graph where V(G)= {a,b,c,d} and E(G) = {[a, b], [b, d], [a, d], [b, c], [c, d]}.

a b

d E

The order of the graph G is 4 and the size of graph Gis 5.

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190. EI GE 102: Mathematics in the Mo X | La M

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A Aron Kim Asprec is presenting

GRAPHS AND GRAPHS MODELS

PICTORIAL REPRESENTATION OF A GRAPH
Let G be a graph where V(G)= (a, b,c, d} and E(G) = {[a, 6], [b, d], [a, d], [b, c], [c, a).

a b
a >
d O

The pictorial representation of a graph is not unique. €

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3 OF 18

GRAPHS AND GRAPHS MODELS

GRAPH
An edge of the form [x, x] is called a loop.

Two edges with the same end points are said to be parallel edges. Two
edges that are incidence on the same endppint are called adjacent edges.
SJ

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A Aron Kim Asprec is presenting

GRAPHS AND GRAPHS MODELS

GRAPH
An edge of the form [x, x] is called a loop.

Two edges with the same end points are said to be parallel edges. Two
edges that are incidence on the same endpoint are called adjacent edges.

A graph without loops and multiple edges is called simple graph.

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e

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A Aron Kim Asprec is presenting

SUBGRAPHS AND DEGREE OF VERTEX

SUBGRAPH
A subgraph can be obtained by removing vertices or edges from a graph.
EXAMPLE: 1 2
GraphG:

Subgraphs of graph G
1 2

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A Aron Kim Asprec is presenting

SUBGRAPHS AND DEGREE OF VERTEX

DEGREE OF AVERTEX

The degree of a vertex x in a graph G is the number of edges incident with x,
denoted by deg(x). —

deg(a) = 2

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A Aron Kim Asprec is presenting

DE 10 OF 18

SUBGRAPHS AND DEGREE OF VERTEX

DEGREE OF A VERTEX

The degree of a vertex x in a graph G is the number of edges incident with x,
denoted by deg(x).

deg(a) = 2
deg(d) =3

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A Aron Kim Asprec is presenting

SUBGRAPHS AND DEGREE OF VERTEX

DEGREE OF AVERTEX

The degree of a vertex x in a graph G is the number of edges incident with x,
denoted by deg(x).

deg(a) =2
deg(d) =3

deg(c) =1
deg(f) = 0

A vertex with degree 0 is called isolated vertex.

M [+] Aron Kim Asprec
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WALK AND CONNECTED GRAPHS

WALK

A walk in a graph is a sequence of vertices such that consecutive vertices
in the sequence are adjacent.

The number of edges in the walk is called length of the walk.

A walk is a closed walk if the first vertex and the last vertex in the
sequence are the same.

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oF 18

WALK AND CONNECTED GRAPHS

WALK

EXAMPLE:

1.The sequence W, =1 4 is a walk oflength 3.

2. The sequence W, d is a walk of length 8.

Specifically, this is a closed walk

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A Aron Kim Asprec is presenting

WALK AND CONNECTED GRAPHS

WALK
EXAMPLE:

1.The sequence W, = 1,2,3,4 is a walk of length 3.
2. The sequence W, = 1,5,3,5,2,1 is a walk of length 5.

Specifically, this is a closed walk.

3. The sequence W; = 5 is a walk of length 0. This is
called the trivial walk.

4. The sequence W, = 1,3,2 is not a walk since 1 and 3
are consecutive vertices in W, and [1,3] is not an edge of
the graph.

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A Aron Kim Asprec is presenting

WALK AND CONNECTED GRAPHS

PATH AND TRAIL

A walk W is called a path if it contains distinct vertices.

AwalkW is called a trail if no edge is repeated.

REMARK:
All paths are trails but not all trails are paths.

VON PATRICK CARAAN has left the meeting

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A Aron Kim Asprec is presenting

DE 14 OF 18

WALK AND CONNECTED GRAPHS

PATH AND TRAIL
EXAMPLE

1. The walk 7,:c,d,f,e is a trail since no edge is
repeated.

2. The walk 7,:f,e, d, f, e is not a trail since an edge is
repeated in the sequence T,.

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AL aron Kim Asprec is presenting

WALK AND CONNECTED GRAPHS

PATH AND TRAIL

EXAMPLE

1. The walk 7,:c,d,f,e is a trail since no edge is
repeated.

2. The walk 72: f,e,d, f, e is not a trail since an edge is
repeated in the sequence T,.

3. The walk P,:b,d,e,f is path since no vertex is
repeated. ra

4. The walk P,:b,d,e,d,c is not a path since vertex d is
repeated in the sequence P,.

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A Aron Kim Asprec is presenting

WALK AND CONNECTED GRAPHS

CIRCUITS AND CYCLES
A closed walk such that no edge is repeated is called circuit.
Aclosed walk that repeats no vertex is called cycle.
——

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Lots

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A Aron Kim Asprec is presenting

WALK AND CONNECTED GRAPHS

CIRCUITS AND CYCLES
EXAMPLE

1. The closed walk C,:c, d, f,e,d,b,c is a circuit since
no edge is repeated.

2.The closed walk C;: b, d, e,d, c, b is not a circuit since
an edge is repeated in the sequence €.

3. The closed walk D;: d, e, f,d is cycle since no vertex
is repeated,

4. The closed walk D,:b,d,e,f,d,c,b is not a cycle
since vertex d is repeated in the sequence D,.

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WALK AND CONNECTED GRAPHS

CONNECTED GRAPH

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ndo?at

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A Aron Kim Asprec is presenting

WALK AND CONNECTED GRAPHS

TRY THIS!
a d

h 1

1. Find a walk with length 4 with k as the first vertex.

2. Find a closed walk with c as the first vertex.
3. List some path and trail with e as the first vertex.

4. List some circuit and cycle with b as the first vertex.
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207. € > (> @ meetgoogle.com/vay-cmrw-ndo?authu!

AL Aron Kim Asprec is presenting

i
1. Find a walk with length 4 with k as the first vertex.

2. Find a closed walk with c as the first vertex.
3. List some path and trail with e as the first vertex.

4. List some circuit and cycle with b as the first vertex.
M [+] Aron Kim Asprec

3:24 PM | vgv-cmrw-ndo