Equivalence classes are either identical or disjoint

Let S be a non-empty set with equivalence relation ~, and ∀a ∈ S let

• [a] = {x∈S | a ~ x}

be the equivalence class of a.

Then two equivalence classes [a] and [b] are either identical or disjoint.

Proof:

Consider [a] ∩ [b]. If [a] ∩ [b] = ∅, well and good. Otherwise, ∃c ∈ S such that c ∈ [a] ∩ [b].

That is, c ~ a and c ~ b. By transitivity of equivalence relations, a ~ b. If x ∈ [a] then x ∈ [b], so [a] ⊆ [b].

Similarly, and by reflexivity of equivalence relations, [b] ⊆ [a].

Since [a] ⊆ [b] and [b] ⊆ [a], [a] = [b].

Thus, equivalence classes are either identical or disjoint.