Let $x \sim y$ be defined as meaning that the ordered tuple $(x, y)$ is in some set $S$. If this is the case, an equivalence relation on the set $S$ is defined as a subset of $S \times S$ with the following properties:
- For all $x$ in $S$, $x \sim x$.
- If $x \sim y$, then $y \sim x$.
- If $x \sim y$ and $y \sim z$, then $x \sim z$.
I think this is an over-determined definition, because (1) is a consequence of (2) and (3), as long as it contains anything in addition to the empty set:
Assume that the equivalence relation contains at least point, $x \sim y$. By (2), it contains $y \sim x$. By (3), since $x \sim y$ and $y \sim x$, we also have $x \sim x$. Since we put no constraints on $x$ except that it be a part of the relation, this implies that for any point $x \sim y$, we have $x \sim x$. (We also have $y \sim y$, since $x \sim y$ implies $y \sim x$.) So why do we need to specify (1)?