such that f ( a ) = g ( a ) ∀ a ∈ A, are called equal functions). . Let A = { , , }. Then number of relations containing ( , ) and ( , ) which are reflexive and symmetric but not transitive is (A) (B) (C) (D) .
Let A = { , , }. Then number of equivalence relations containing ( , ) is (A) (B) (C) (D) Summary In this chapter, we studied different types of relations and equivalence relation, composition of functions, invertible functions and binary operations. The main features of this chapter are as follows: Empty relation is the relation R in X given by R = φ ⊂ X × X. Universal relation is the relation R in X given by R = X × X.
Reflexive relation R in X is a relation with ( a , a ) ∈ R ∀ a ∈ X. Symmetric relation R in X is a relation satisfying ( a , b ) ∈ R implies ( b , a ) ∈ R. Transitive relation R in X is a relation satisfying ( a , b ) ∈ R and ( b , c ) ∈ R implies that ( a , c ) ∈ R. Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.
Equivalence class [ a ] containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a . A function f : X → Y is one-one (or injective ) if f ( x ) = f ( x ) ⇒ x = x ∀ x , x ∈ X. A function f : X → Y is onto (or surjective ) if given any y ∈ Y, ∃ x ∈ X such that f ( x ) = y . A function f : X → Y is one-one and onto (or bijective ), if f is both one-one and onto.