= Solution
A <binary relation> $R$ on $A$ is a <subset> of $A\times A$. Its properties are quantified over elements of $A$:
* A <reflexive relation> satisfies $\forall a\in A,\ aRa$.
* A <symmetric relation> satisfies $\forall a,b\in A,\ aRb\Rightarrow bRa$.
* An <antisymmetric relation> satisfies $\forall a,b\in A,\ (aRb\land bRa)\Rightarrow a=b$.
* A <transitive relation> satisfies $\forall a,b,c\in A,\ (aRb\land bRc)\Rightarrow aRc$.
* An <irreflexive relation> satisfies $\forall a\in A,\ \neg(aRa)$.
* A <serial relation> satisfies $\forall a\in A\ \exists b\in A,\ aRb$.
Thus an <equivalence relation> is a <binary relation> that is simultaneously reflexive, symmetric and transitive. The <equivalence class> of $a$ is the <set> of all elements related to it:
$$
\boxed{R\text{ is an equivalence relation}\iff R\text{ is reflexive, symmetric and transitive};\qquad[a]_R=\{b\in A:aRb\}.}
$$
<Antisymmetry> permits diagonal pairs and does not mean that a <binary relation> fails to be a <symmetric relation>.
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