Prove that a relation that is reflexive and symmetric need not also be transitive.

We give a counterexample:

Let R={ (a, a), (a, b), (b, a), (b, b), (c, c), (b, c), (c, b) } be a relation.

It is reflexive as it contains (a, a), (b, b) and (c, c).

It is symmetric as (a, b) and (b, a) are both in R; (b, c) and (c, b) are both in R.

But it is not transitive, as (a, b) and (b, c) are in R, but (a, c) is not.