For relations that are not equivalence relations, what does it mean to calculate the smallest equivalence relation that contains it?

Question

One set in question: S = {(a,a), (a,b), (b,b), (c,b), (c,c)}.

Answer 1

The set has to be reflexive, transitive and symmetric. So the above set is already reflexive and transitive. Reflexive means that if $a$ is an element in one of the pair in the relation, then the pair $(a,a)$ is also in the relation. Transitive means that if $(a,b) and (b,c)$ is in the relation, then $(a,c)$ is n the relation. We just need to make sure that it is symmetric. Symmetric means that if $(a,b)$ is in the relation, then $(b,a)$ is also in the relation. In the given relation, $(a,b)$ did not have its equivalent $(b,a)$ and $(c,b)$ did not have its equivalent $(b,c)$. So add $\{((b,a),(b,c)\}$ to get $\{(a,a),(a,b),(b,b),(c,b),(c,c),(b,a),(b,c)\}$. The right way to think of an equivalence relation is as a partition of the set, where two elements are in the same partition if the relation relates them. Which means every partition of the set defines an equivalence relation on the set. And so the set of all equivalent relation on a set is just the possible ways you can partition a set.

Answer 2

Hint: Its the reflexive, symmetric and transitive hull of the given relation $S$, i.e., the smallest relation containing $S$ which is reflexive, symmetric and transitive.