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 is S.

- Quotient set: The collection of all equivalence classes is called the quotient set, written S/~. Each element of S belongs to one class, and the classes provide a reduced description of S under the relation ~.

- Notation: [a] is the standard notation for the class of a. Some contexts use a ~ b to indicate membership in the same class without listing all members.

Examples

- Modular congruence: On the integers Z, define a ~ b if a ≡ b (mod n). Then [a] is the set of all integers congruent to a modulo n; there are exactly n distinct classes in Z/~.

- Equality relation: If ~ is equality (a ~ b iff a = b), each class [a] is the singleton {a}.

Applications

Equivalence classes are central to forming quotient structures in algebra (quotient groups, rings, modules), topology (quotient spaces), and various areas of combinatorics and logic. They provide a rigorous way to identify elements that share a specified property and to study the resulting simplified structure.