Equivalence class - Knowlpedia

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 .

An equivalence class of an element $a\in A$ (with respect to an equivalence relation $\sim$ on $A$ ) is the set

[a]=\{x\in A : x\sim a\}.

Remarks

Equivalence classes are subsets of $A$ that bundle together elements deemed “the same” by $\sim$ . The collection of all equivalence classes is the quotient set $A/{\sim}=\{[a]: a\in A\}$ , and it is a partition of $A$ .

Examples

If $\sim$ is congruence modulo $3$ on $\mathbb{Z}$ , then $[1]=\{\dots,-5,-2,1,4,7,\dots\}$ .
If $\sim$ is equality on a set $A$ , then for each $a\in A$ one has $[a]=\{a\}$ .