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

The **Answer to the Question**

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**Here's the Solution to this Question**

Given relation is-

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

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

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

Example:Β $(2, 4) β R βΉ (4, 2) β R.$

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

Example:Β $(3, 1) β R \text{ and } (1, 3) β R βΉ (3, 3) β R.$

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