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Give a specific reason why the following set R does not define an equivalence relation on the set {1, 2, 3, 4}.R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 3), (3, 2), (2, 4), (4, 2)}



Answer :

The following set R does not define an equivalence relation, as it is not satisfying the transitive property.

Equivalence relation: In mathematics equivalence relation is a kind of binary relation which is reflexive, symmetric, and transitive.

The given set is {1, 2, 3, 4} which is forming a relation R where R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 3), (3, 2), (2, 4), (4, 2)}

This relation is following reflexive property as (1, 1), (2, 2), (3, 3), (4, 4) all are present in the relation

This relation is following symmetric property as  (2, 3), (3, 2), (2, 4), (4, 2) all are there in the relation

But the relation is not following transitive property as (2,3) ,(4,2) are there but (4,3) is not there in the relation . So it is not transitive.

Therefore the given relation is not an equivalence relation because it is not transitive .

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