Module 3. mathematics in the modern world

basic concepts of math

1. MODULE-3 (FOUR BASIC CONCEPTS:SETS,RELATIONS, FUNCTIONS,AND BINARY OPERATIONS)

2. The language of Sets What is set? -According to Cantor a set is a well-defined collection of distinct objects.The term “well defined” means that we can tell whether a certain objects is a member of the collection or not. Example of Sets: •a,e,i,o,u which is the set of letters •1,2,5 and 9 which is the set of numbers •The Sets of days of the week that begin with letter T •The Sets of letters of the English alphabet •The Sets of odd whole positive numbers •The Sets of UEP students •The Sets of negative integers

3. There are two ways of representing sets: Roster Method -Defined as a way to show the elements of set by listing the elements inside of brackets. Example: -Set of all vowels in the English alphabet:{a,e,i,o,u}. -Set of all odd positive integers less than 10:{1,3,5,7,9} ° Set –builder notation - Is a mathematical notation of describing a set by representing it’s elements or explaining the properties that’s it’s members must satisfy . Example: 1.The set of letters a,e,i,o,u •A= {x|x is a vowel} read as “ A is the set of all elements such that x is a vowel”

4. What is interval? -An interval is a set of real numbers that has two endpoints, endpoints are not included {X € R|a <x<b} -An interval is a set of real numbers that has two endpoints, only left-handed endpoint is included { X€ R|a ≤ x < b} -An interval is a set of real numbers that has two endpoints ,only Right –handed endpoint is included {X € R|a < x ≤ b } -An interval is a set of real numbers that has two endpoints both endpoints included {X € R|a ≤ x ≤ b) ° -An Interval is a set that has one endpoint, endpoint and every real number to it’s right is included {x € R|a ≤ x}

5. A set can be Part of another Set Subset -Set A is a subset of B written, A ⊆ B if and only if each element of set A is an element of Set B. That is ,if x A, then x B. Set A is contained in set B. °Proper Subset -Set A is a Proper subset of B, written A ⊂ B if and only if each element of set A is an element of set B but there is at least one element in B that is not contained in A. Example: 1.B is a proper subset of B -Since B = B so it is false.  

6. Ordered pair -Given the elements a and b,the ordered pair consisting these elements has the specification that a is the first element and b is the second element. In Symbols,( a,b ) Two ordered pairs (a, b) and (c, d) are equal if a=c and b=d °Cartesian product - Given sets A and B, the cartesian product of A and B, is the set of all ordered pairs ( a,b ) ,where a A and b B. In symbols, A x B = { ( a,b )|a A and b B} A x B is read as “A cross B”. Example: 1.Let A= { a, b} and B = { 1,2,3} ,Find :A x B Solution: A x B = {(a,1),(a,2),(a,3),(b,1) ,(b,2),(b,3)}  

7. The language of Relations and functions Relations - Expresses an association between two objects. -Let A and B be sets A relation R from A to B is a subset of A x B .Given an ordered pair (x , y) in A x B, x is related to y by R, written x R y if and only if. (x , y) is in R. •When x is related to y by R: x R y means that (x , y) R ° The Domain and the range of the Relation - Let R be a relation from set A into set B. Then the domain not R denoted by D(R) is the set D(R)= { a | a A and there exist b B such that ( a,b ) R}.  

8. The range on image of R denoted by Im (R),is the set Im (R) ={b | b B and there exist a A such that ( a,b ) R ). It follows that: D (R) is the set of all elements of A that are related to some elements of B. Im (R) is the set of all those elements of B that have some element of A related to them .  

9. FUNCTIONS A function is a special relationship between two sets, where each input (from the domain) is paired with exactly one output (from the range). Mathematical Definition: A function is a rule that assigns each element of a set x (the domain) to a single element of a set y (the range). Notation: Functions are often written as f(x), where: x is the input (independent variable) f(x) is the output (dependent variable) The function rule tells us how to get from x to f(x).

10. Function vs. Non-Function A relation is a function if each input has exactly one output . 🔹 Example of a Function: f(x)= If x=2, then f(2)= = 4 If x=−2, then f(−2)=( = 4. Every input gives only one output, so this is a function. 🔹 Example of a Non-Function: A relationship where an input leads to multiple outputs is not a function. For example: = x If x=4, then y could be 2 or −2, meaning one input gives two outputs. Therefore, this is not a function .  

11. Ways to Represent a Function Functions can be represented in different ways: ✅ Verbal Representation (Word Description) "The amount of money you earn is a function of the number of hours you work." ✅ Table Representation x (Input) f(x) (Output) 1 2 2 4 3 6 ✅ Graph Representation A function is a curve or line where each x-value has only one y-value . Use the Vertical Line Test : If a vertical line passes through more than one point, it's not a function. ✅ Equation Representation Example: f(x)=2x+3. This function takes x, doubles it, and adds 3 to get the output.

12. Types of Functions Functions come in different forms, depending on how they behave: 🔹 Linear Function: Form: f(x)=mx + b Example: f(x) = 2x+3 (Straight-line graph) 🔹 Quadratic Function: Form: f(x)=ax²+bx+c Example: f(x)=x²−4x+3 (Parabolic shape) 🔹 Exponential Function: Form: f(x)= Example: f(x)= (Used in population growth, interest rates, etc.) 🔹 Piecewise Function: Different rules for different values of x. Example: A delivery fee that is $5 for orders below $50 and free for orders above $50.  

13. Functions -Let A an be non-empty sets and f be Relation from A into B.Then f is called a function from A into B if i .the domain of f is A,that is D ( f) = A ,and ii. For all a € A ,b,€ B, and c € b if ( a,b ) € f and ( a,c ) € f,then b = c. °Binary Functions - Let S be a non – empty set .A binary operation on S is a function from S x S into S °Some properties of Binary Operation -Let S be a non empty set and* a binary operation on S.Then i.* is called associative It for all x , y,z € S, x* (y * z) = x * y * z ii.* Is called commutative if for all x,y € S, x * y = y* x