Quotient set - Imported Knowlpedia Content

Reflexive: a ∼ a a\sim a a ∼ a .
Symmetric: if a ∼ b a\sim b a ∼ b , then b ∼ a b\sim a b ∼ a .
Transitive: if a ∼ b a\sim b a ∼ b and b ∼ c b\sim c b ∼ c , then a ∼ c a\sim c a ∼ c .

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

A/{\sim}=\{[a]_{\sim}: a\in A\},\qquad [a]_{\sim}=\{x\in A: x\sim a\}.

Remarks

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.