A relation $R$ between two non-empty sets $S$ and $T$ is a subset of the Cartesian product $S\times T$.
We say that $s$ is related to $t$ if and only if $(s,t)\in R$, that we also write as $sRt$.
In Abstract Algebra, many important relations $R$ have $S=T$ so that, in particular:
$$R\subseteq S\times S$$
Such a relation is called:
Reflexive if $(s,s)\in S\times S$ for all $s\in S$;
Symmetric if $(s,t)\in R$ implies $(t,s)\in R$;
Transitive if $(s,t)\in R$ and $(t,u)\in R$ implies $(s,u)\in R$.
A relation that is reflexive and symmetric and transitive is called an equivalence relation and these are very important for Abstract Algebra.
When $R\subseteq S\times S$ is an equivalence relation, we usually change notation and write $s\sim t$ for $(s,t)\in R$ and for $sRt$.
We like equivalence relations because the $\sim$ shares many properties with the usual $=$ sign, and it can be very useful to view $s$, $t$ in $S$ with $s\sim t$ as being the same.
Which of the following $R\subseteq S\times S$ defines an equivalence relation on $S$?