Quotient set

A quotient set is the set of equivalence classes determined by an equivalence relation: if $A$ is a set and $\sim$ is an equivalence relation on $A$, then the quotient set $A/{\sim}$ is

Each element of $A/{\sim}$ is an equivalence class , and the collection of all classes forms a partition of $A$. Conversely, any partition of $A$ determines an equivalence relation and hence a quotient set.

Examples:

• On the integers , fix $n\in\mathbb{N}$ with $n\ge 2$ and define $a\sim b$ if $a-b$ is divisible by $n$; then $\mathbb{Z}/{\sim}$ is the set of congruence classes modulo $n$.
• On the real numbers , define $x\sim y$ if $x-y\in\mathbb{Z}$; then $\mathbb{R}/{\sim}$ identifies real numbers that differ by an integer.