This is one of the problem I have been solving in Velleman's How to Prove book:
Suppose $R$ is a relation on $A$, and let $S$ be the transitive closure of $R$. Prove that if $R$ is symmetric, then so is $S$. (Hint: Asumme that $R$ is symmetric. Prove that $R \subseteq S^{-1}$ and $S^{-1}$ is transitive. What can you conclude about $S$ and $S^{-1}$?)
Now I have proved both $R \subseteq S^{-1}$ and $S^{-1}$ is transitive. Now, what I cannot figure out is the relationship between $S$ and $S^{-1}$. Is there any general method for determining it ?
(Note: I haven't been taught Induction yet in the book, so I would like to stay away from it if there is any other way of determining the relationship between $S$ and $S^{-1}$)
Hint: work an example. For instance, on the set $\{1, 2, 3, 4, 5, 6, 7\}$, let $R$ be the relation $$R(a, b)\iff \vert a-b\vert=2.$$ $R$ is symmetric; compute $S$ and $S^{-1}$ and compare them. Now make a guess . . .