What is equivalence relation? Explain use of equivalence relation with the help of an example.

Equivalence Relations: A relation R on a set A is called an equivalence relation if and only if

(i) R is reflexive, i.e., for all a ∈ R, (a, a) ∈ R,

(ii) R is symmetric, i.e., (a, b) ∈ R ⇒ (b, a) ∈ R, for all a, b ∈ A, and

(iii) R is transitive, i.e., (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R, for all a, b, c ∈ A.

One of the most trivial examples of an equivalence relation is that of ‘equality’.

For any elements a,b,c in a set A,

(i) a = a, i.e., reflexivity

(ii) a = b⇒ b = a, i.e., symmetricity

(iii) a = b and b = c ⇒ a = c , i.e., transitivity.