Verify whether the given binary relation R on the set is an

Verify whether the given binary relation R on the set is an equivalence relation: Let A = {1, 2, 3, 4, 5} and let R be the binary relation whose elements are the pairs (1, 1), (1, 3), (1, 4), (2, 2), (2, 5), (3, 1), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4), (5, 2), (5, 5), and only these. If you found an affirmative answer in l.(a), then find quotient set A/R.

Solution

For equivalence relation, we must have a relation reflexive, symmetric and transitive all.

So for reflexive : aRa or (a,a) should be in R

and in given relation it is as (1,1),(2,2),(3,3), (4,4) and (5,5) are in R, so it is reflexive.

Further for symmetry if aRb then bRa

And pairs (1,3),(3,1),(2,5),(5,2) etc, show that symmetry is there.

Further for transitive, we have if aRb and bRc, then aR c

But here we do not find any such relation of three pairs that follow above rule, so it is not transitive and hence it is not equivalence.