Question

Check whether the relation R on the set S = {1, 2, 3} is an equivalent relation where:                                           

 𝑅 = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has: reflexive, symmetric, anti-symmetric,  transitive? Justify your answer in each case?

Answer 1

Given relation is-

 R = {(1, 1), (1, 3), (2, 2), (2, 4), (3, 1), (3, 3), (4, 2), (4, 4)}.

Reflexive: Relation R is reflexive as $(1, 1), (2, 2), (3, 3) \text{ and } (4, 4) ∈ R.$

Symmetric: Relation R is symmetric because whenever (a, b) ∈ R, (b, a) also belongs to R.

Example: $(2, 4) ∈ R ⟹ (4, 2) ∈ R.$

Transitive: Relation R is transitive because whenever (a, b) and (b, c) belongs to R, (a, c) also belongs to R.

Example: $(3, 1) ∈ R \text{ and } (1, 3) ∈ R ⟹ (3, 3) ∈ R.$

So, as R is reflexive, symmetric and transitive, hence, R is an Equivalence Relation.