An equivalence relation $\sim$ on a set $S$ is a relation that is:

Reflexive.

$a\sim a$ for all $a\in S$.

Symmetric.

Whenever $a\sim b$, then $b\sim a$.

Transitive.

If $a\sim b$ and $b\sim c$ then $a\sim c$.

If $a$ and $b$ are related this way we say that they are equivalent under $\sim$. If $a\in S$, then the set of all elements of $S$ that are equivalent to $a$ is called the equivalence class of $a$. The set of all equivalence classes under $\sim$ is written $S/\sim$.

An equivalence relation on a set induces a partition on it. Conversely, any partition induces an equivalence relation. Equivalence relations are important, because often the set $S$ can be ’transformed’ into another set (quotient space) by considering each equivalence class as a single unit.

Two examples of equivalence relations:

1. Consider the set of integers $\mathbbmss{Z}$ and take a positive integer $m$. Then $m$ induces an equivalence relation by $a\sim b$ when $m$ divides $b-a$ (that is, $a$ and $b$ leave the same remainder when divided by $m$).

2. Take a group $(G,\cdot)$ and a subgroup $H$. Define $a\sim b$ whenever $ab^{{-1}}\in H$. That defines an equivalence relation. Here equivalence classes are called cosets.