Transitive, symmetric $R$ such that for all $x$, there is $y$ such that $xRy$, is an equivalence relation

Question

I'm stuck at one particular task I'm working on.

Here is the task:

Let R be a transitive and symmetrical relation on $S$. Assume that for all $x \in S$ there is a $y \in S$ so that $xRy$. Prove that $R$ is an equivalence relation.

How can I prove that $R$ is an equivalence relation?

I would appreciate any help.

Thank you.

Answer 1

All you need to do here is make the case that $R$ must be reflexive. Since for every $a \in S$ there is $b,$ such that $aRb,$ then by symmetry $bRa$ and by transitivity $aRa.$ Since this holds for every $a\in S$, the relation is necessarily reflexive, and hence, is an equivalence relation.

Answer 2

We know $xRy$. Since it is symmetric $yRx$. Thus $xRy$ and $yRx$ so by transitivity $xRx$. Thus the relation is also reflexive show that it is an equivalence relation.