Relation and function
- 1. RELATIONS AND FUNCTIONS Made by:-Aaditya Gera Class :-12th-A Roll no.:-1
- 2. What is a RELATION? A connection between the elements of two or more sets is Relation .Let A and B be two sets such that A = {2, 5, 7, 8, 10, 13} and B = {1, 2, 3, 4, 5}. Then, R = {(x, y): x = 4y – 3, x ∈ A and y ∈ B} (Set-builder form) R = {(5, 2), (10, 3), (13, 4)} (Roster form)
- 3. TYPES OF RELATION:- 1) UNIVERSAL 2) IDENTTY 3) SYMMETRIC 4) INVERSE 5) REFLEXIVE 6) TRANSITIVE 7) EQUIVALENCE
- 4. RELATIONS Universal Relation • A relation R in a set, say A is a universal relation if each element of A is related to every element of A, i.e., R = A × A. Also called Full relation. Suppose A is a set of all natural numbers and B is a set of all whole numbers. The relation between A and B is universal as every element of A is in set B. Identity Relation • In Identity relation, every element of set A is related to itself only. I = {(a, a), ∈ A}. For example, If we throw two dice, we get 36 possible outcomes, (1, 1), (1, 2), … , (6, 6). If we define a relation as R: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, it is an identity relation.
- 5. RELATIONS Symmetric Relation • A relation R on a set A is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A. Inverse Relation • Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R-1 is said to be an Inverse relation if R-1 from set B to A is denoted by R-1 = {(b, a): (a, b) ∈ R}. Considering the case of throwing of two dice if R = {(1, 2), (2, 3)}, R-1 = {(2, 1), (3, 2)}. Here, the domain of R is the range of R-1 and vice-versa.
- 6. RELATIONS Reflexive Relation • If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R. Transitive Relation • A relation in a set A is transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A Equivalence Relation • A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive.
- 7. WHAT IS A FUNCTION? • A function is a binary relation over two sets that associates every element of the first set, to exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.
- 8. TYPES OF FUNCTIONS 1) ONE-ONE(INJECTIVE)FUNCTION 2) ONTO(SURJECTIVE)FUNCTIONS 3) ONE-ONE AND ONTO(BIJECTIVE)FUNCTIONS
- 9. ONE-ONE(INJECTIVE)FUNCTION • In this function every element of the function's codomain is the image of at most one element of its domain. • Let f be a function whose domain is a set X. The function f is said to be injective provided that for all a and b in X, whenever f(a) = f(b), then a = b; that is, f(a) = f(b) implies a = b. Equivalently, if a ≠ b, then f(a) ≠ f(b).
- 10. ONTO(SURJECTIVE)FUNCTIONS • A surjective function is a function whose image is equal to its codomain ,if for every element y in the codomain Y of f, there is at least one element x in the domain X of f such that f(x) = y.
- 11. ONE-ONE AND ONTO(BIJECTIVE)FUNCTIONS • Bijective is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements.
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