ekvivalenciareláció

Ekvivalenciareláció (equivalence relation) is a binary relation on a set that satisfies three fundamental properties: reflexivity, symmetry, and transitivity. Reflexivity means every element of the set is related to itself. Symmetry requires that if an element a is related to an element b, then b is also related to a. Transitivity demands that if a is related to b and b is related to c, then a must be related to c. Only relations that meet all three conditions qualify as equivalence relations.

The primary consequence of an equivalence relation is that it partitions the underlying set into disjoint equivalence classes. Each class contains elements that are mutually related, and no element from one class relates to any element in another. The set of all such classes forms a quotient set, often denoted by the original set modulo the relation, for example, ℤ/∼ for integers modulo an equivalence relation.

Common examples include congruence modulo an integer n, where two integers have the same remainder when divided by n; identity relation, where only identical elements are related; and similarity of geometric figures, where shapes can be transformed into one another by rigid motions. Equivalence relations arise naturally in mathematics, enabling the construction of quotient structures in group theory, topology, algebraic geometry, and computer science. They provide a systematic way to identify and handle objects that are considered the same under a specified criterion.