Is it possible for a function to be an equivalence relation?

Question

I am not quite sure about my arguments:

1. $aRa$ for every pair $(a,a)$ in $S\times S$, where $S$ — some abstract set. Therefore, every value of a function must be the same as it's argument — $f(x) = x$, for every $x$ in $S$.

2. $aRb \implies bRa$. This will work if our function is the one described above.

3. $aRb, bRc \implies aRc$. Will work also for $f(x) = x$, in case of $a = b = c$.

Is it correct?