Equivalence classes

For an equivalence relation R on X and a ∈ X, the equivalence class [a] is the set of all elements of X related to a.

Under a R b ⇔ 2 divides a − b on Z, every even integer is related to 0 and no odd one is, so [0] = E, the even integers, and [1] = O, the odd integers. E and O are disjoint and E ∪ O = Z; also [0] = [2r] and [1] = [2r + 1] for every r ∈ Z.

In general an equivalence relation cuts X into mutually disjoint subsets Aᵢ, the equivalence classes, such that any two members of the same Aᵢ are related, members of different parts Aᵢ and Aⱼ (i ≠ j) never are, and together the parts make up all of X.

The process runs backwards too: split Z into A₁ = {…, −3, 0, 3, 6, …}, A₂ = {…, −2, 1, 4, 7, …} and A₃ = {…, −1, 2, 5, 8, …}, the integers leaving remainder 0, 1 and 2 on division by 3. Calling two integers related when they sit in the same part gives a R b ⇔ 3 divides a − b, with A₁ = [0], A₂ = [1], A₃ = [2].

Any two classes are either identical or disjoint: [a] = [b] exactly when a R b. For divisibility of a − b by 3, [3r] = [0], [3r + 1] = [1] and [3r + 2] = [2].

On {1, 2, 3, 4, 5, 6, 7}, 'a and b are both odd or both even' has exactly two classes, {1, 3, 5, 7} and {2, 4, 6}; every element of one is related to every other element of it and to nothing in the other.

For points of a plane related when they are the same distance from the origin, the class of a point P other than the origin is the circle through P centred at the origin.