Let E be an equivalence relation on a set X. Recall that for each x ∈ X, the E-equivalence class of x is the set [x]E := {y ∈ X : xEy} The following outlines a proof that the E-equivalence classes partition X. (a) Prove: For all x, y ∈ X, if xEy then [y]E ⊆ [x]E. (b) Deduce: For all x, y ∈ X, if xEy then [y]E = [x]E. (c) Deduce: For all x, y, z ∈ X, if z ∈ [x]E ∩ [y]E then [x]E = [z]E = [y]E. (d) Conclude: The set {[x]E : x ∈ X} of E-equivalence classes is a partition of X, i.e., X is a disjoint union of E-equivalence classes (A disjoint union is just a union of pairwise disjoint sets.).

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