Let $R$ be a binary relation on a set A and suppose R is symmetric and transitive. Prove the following: If for every $x$ in $A$ there is a $y$ in $A$ such that $x R y$, then $R$ is an equivalence relation.
I have no idea where to begin. Please, can someone provide a hint or a guide of where to begin? Thank you so much.
HINT: The one additional property that $R$ needs in order to be an equivalence relation is reflexivity. Thus, you have to prove that $x\mathrel{R}x$ for each $x\in A$. You’ll need to use all three of the given properties: that there is a $y\in A$ such that $x\mathrel{R}y$, that $R$ is symmetric, and that $R$ is transitive. In fact, you’ll need to use them in exactly that order — and that’s a fairly broad hint right there.