Let $S$ be a set. An equivalence relation on $S$ is a subset $R\subseteq S\times S$ satisfying that $(s,s)\in R$ for all $s\in S$, $(s,t)\in R \implies (t,s)\in R$, $(s,t)\in R\; \land\; (t,u)\in R\implies (s,u)\in R$.
My question is: What is the definition of a relation?
Is a relation just any subset of $S\times S$
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$\begingroup$Yes, a relation on $S$ is just a subset of $S\times S$, and any subset of $S\times S$ is a relation on $S$.
It comes with a notational quirk, though. If $R$ is such a relation, we may write $sRt$ for $(s,t)\in R$. You commonly see this in expressions like $2\leq3$ (no mathematician in their right mind would seriously prefer to write $(2,3)\in {}\leq$ unless they, like I am here, are making some very specific point).
$\endgroup$ $\begingroup$More generally a relation can defined as a binary predicate, that's a statement with two free variables $\mathrm R_{x,y} $. For example, $x\in y $ is a relation respect to the free variables $x $ and $y $.
Now, given a relation $\mathrm R_{x,y} $, its graph is the class$$R=\{(x,y):\mathrm R_{x,y}\} $$Moreover, given a class (or a set) $X $, the graph of $\mathrm R_{x,y} $ restricted to $S$ is$$R=\{(x,y)\in S\times S:\mathrm R_{x,y}\} $$
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