Does a symmetric closure preserve transitivity?

Question

My intuition tells me that it does. As simply adding an edge in the opposite direction of an existing edge won't impact a series of nodes such that

$aRb \wedge bRc \rightarrow aRc$

But how could I prove this?

Answer 1

Let $R=\{(a,b),(c,b)\}$, a transitive relation on $\{a,b,c\}$. Let $S$ be the symmetric closure of $R$. Then $aSb\wedge bSc$ but not $aSc$.