Lesson 5. Relations- Functions- and Binary Operations.pdf

Republic of the Philippines BICOL STATE COLLEGE OF APPLIED SCIENCES AND TECHNOLOGY City of Naga MATHEMATICS IN THE MODERN WORLD LESSON 5. RELATIONS, FUNCTIONS, AND BINARY O...

Republic of the Philippines BICOL STATE COLLEGE OF APPLIED SCIENCES AND TECHNOLOGY City of Naga MATHEMATICS IN THE MODERN WORLD LESSON 5. RELATIONS, FUNCTIONS, AND BINARY OPERATIONS Relations 4. y2=x -A relation from set X to Y is the set of ordered pairs of real 5. x2+y2=1 numbers (x, y) such that to each element x of the set X there 6. corresponds at least one element of the set Y. Example: A = {1, 2, 3} and B = {2, 3, 4} Condition to be related: an element x in A is related to an element y in B if and only if, x is less than y. Operations on Functions Exercise 1 Let A = {1,2} and B = {1,2,3} *Addition and Subtraction: (f ± g)(x) = f(x) ± g(x) a. Determine if the elements of A x B is R *Multiplication: (f·g)(x) = f(x)·g(x) b. Is 1 R 3? Is 2 R 3? Is 2 R 2? *Division: f(x)/g(x), where g(x) is not equal to zero c. What are the domain and the co-domain of R? *Composition: (f ο g)(x) = f(g(x)) Arrow Diagram of a Relation Unary VS Binary Operation Given a relation {(1, 2),(0, 1),(3, 4),(2, 1),(0, −2)}. Illustrate the given relation into an arrow diagram. Properties of Binary Operations 1. Closure Property Exercise 2 2. Commutative Property Given A = {1,2,3} and B = {1,3,5}. Let S be the set of related (x,y). Draw the arrow diagram of S. 3. Associative Property Function Relation A function is a relation in which every input is paired with exactly one output. 4. Distributive Property A function from set X to Y is the set of ordered pairs of real numbers (x, y) in which no two distinct ordered pairs have the same first component. 5. Identity Elements Exercise 3 Determine if the ff is a function relation. 1. {(0, -5), (1, -4), (2, -3), (3, -2), (4, -1), (5, 0)} 6. Inverses 2. 2x + 3y – 1 = 0 3. y=x2 Addition (0), Multiplication (1) 1  Republic of the Philippines BICOL STATE COLLEGE OF APPLIED SCIENCES AND TECHNOLOGY City of Naga MATHEMATICS IN THE MODERN WORLD Application of Binary Operation Example: 1. If a * b = 3a – 2b + ab, find 4*1. 2. If a * b = a2– b, find the values of a such that a * 4 = 5 3. If a - b = 2a + b and a * b = a –2b, find (3 - 4) * 5. Solution: 2