equivalenssiluokan
Equivalenssiluokan, or equivalence class, is a fundamental concept in mathematics used to group elements that are considered equivalent under a given relation. Suppose we have a set S and an equivalence relation ~ on S. The equivalence class of an element a ∈ S is the subset [a] = { x ∈ S | x ~ a }. The relation must be reflexive, symmetric, and transitive, which ensures that every element belongs to exactly one class and that classes partition S.
Key properties and constructions
- Partition: The equivalence classes form a partition of S; they are pairwise disjoint and their union
- Quotient set: The collection of all equivalence classes is called the quotient set, written S/~. Each
- Notation: [a] is the standard notation for the class of a. Some contexts use a ~ b to
- Modular congruence: On the integers Z, define a ~ b if a ≡ b (mod n). Then [a]
- Equality relation: If ~ is equality (a ~ b iff a = b), each class [a] is the singleton
Equivalence classes are central to forming quotient structures in algebra (quotient groups, rings, modules), topology (quotient