Equivalensklass

Equivalensklass, or equivalence class, is a fundamental concept in mathematics. Given a set A and an equivalence relation ~ on A, the equivalence class of an element a ∈ A is the subset [a] = { x ∈ A | x ~ a }. An equivalence relation is a relation that is reflexive, symmetric and transitive.

Equivalence classes partition the set A. This means every element of A belongs to exactly one equivalence class, and if a ~ b then [a] = [b]. The collection of all equivalence classes is called the quotient set, often written as A/~ or A modulo ~. There is a natural map π: A → A/~ defined by π(a) = [a], called the canonical projection.

In practice, equivalence classes are used to identify elements that are considered the same under the chosen relation, allowing the construction of a simplified or "quotient" structure. Operations on A/~ are defined only when the underlying structure on A is compatible with the equivalence relation.

Examples help illustrate the idea. In modular arithmetic, two integers a and b are equivalent modulo n if a − b is a multiple of n; the equivalence classes are the residue classes [0], [1], ..., [n−1]. More generally, a ~ b if they have the same remainder when divided by n. Another example is strings being equivalent if they are anagrams of each other; the equivalence class of a word consists of all its anagrams.

Equivalensklass provides a formal way to group objects by a shared property and to form quotient structures in algebra, topology, and beyond.