equivalenceklasse

An equivalence class is a subset of a set S determined by an equivalence relation ~ on S. For any element a in S, the equivalence class of a is [a] = { x in S | x ~ a }. An equivalence relation on S is a relation that is reflexive, symmetric, and transitive; these three properties guarantee that the elements of S can be partitioned into disjoint classes.

The collection of all equivalence classes forms the quotient set S/~, and the natural map π: S -> S/~ sending x to [x] is surjective. Every element of S lies in exactly one equivalence class; the classes are pairwise disjoint and their union is S.

Common examples include integers with congruence modulo n: x ~ y iff x − y is divisible by n; then [a] = a + nZ. Another example is the relation on R defined by floor(x) = floor(y), which partitions R into the unit intervals [k, k+1). More generally, any partition of S into nonempty, pairwise disjoint subsets determines an equivalence relation by x ~ y iff x and y lie in the same block of the partition.

Equivalence classes are fundamental in algebra, geometry, and topology because they allow the construction of quotient structures, such as quotient groups, rings, and topological spaces, by identifying elements that are related. In practice, one often works with representatives of classes, or with the quotient object S/~ loaded with the induced structure.