1.1Equivalence Relations

An equivalence relation on a set $A$ is any relation $\sim$ satisfying the three following properties:

1. Reflexivity. $a \sim a$ for every $a \in A$.

2. Symmetry. $a \sim b$ if and only if $b \sim a$ for every $a, b \in A$.

3. Transitivity. If $a \sim b$ and $b \sim c$, then $a \sim c$ for every $a, b, c \in A$.

Equivalence relations partition a set $S$, where a partition of a set $S$ is a set of nonempty, disjoint subsets of $S$ whose union is $S$ itself. For partitions induced by equivalence relations, we say each subset is an equivalence class, defined by

The quotient of a set $S$ with respect to an equivalence relation $\sim$ is the set of all equivalence classes induced by $\sim$. That is,

Note. Sets are deduplicated by definition, so redundant equivalence classes are culled.

Take $S = \bZ$ and $\sim$ to be such that

Then $\bZ/{\sim} = \{[0]_\sim, [1]_\sim\}$, with

A word is in order about equivalence classes of numbers modulo $n$, which will be discussed in the near future.