Relations

A relation on a set $S$ is a set of ordered pairs of elements from $S$. Therefore, the relation is a subset of $S\times S$. An equivalence relation on a set $S$ is a relation $R$ satisfying certain properties. Write $xRy$ to mean $(x,y)$ is an element of $R$, and we say $x$ is related to $y$, then the properties are

Reflexive:

$xRx$ for all $x \in S$,

Symmetric:

$xRy$ implies $yRx$ for all $x \in S$ and $y \in S$.

Transitive:

$xRy$ and $yRz$ imply $xRz$ for all $x \in S$, $y \in S$ and $z \in S$.

Where these three properties are completely independent. Other notations are also often used to indicate a relation, e.g., $a \equiv b$ or $a \sim b$. Good reference for set theory: http://mathworld.wolfram.com/Set.html