ekvivalens

Ekvivalens, or equivalence, is a foundational concept used in mathematics, logic, and related disciplines. It describes a relation that groups objects into classes of elements that are considered alike under a given notion of sameness. In formal terms, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive.

A relation ~ on a set S is reflexive if every element a in S satisfies a ~ a. It is symmetric if a ~ b implies b ~ a for any a, b in S. It is transitive if a ~ b and b ~ c imply a ~ c for any a, b, c in S. When these three properties hold, the set S can be decomposed into equivalence classes, where the class of an element a is the set [a] = { x in S | x ~ a }. The collection of all such classes is called the quotient set S/~.

Equivalence relations arise in many contexts. Equality between objects is the simplest example, since a = b is reflexive, symmetric, and transitive. Other examples include congruence modulo n on integers (a ~ b if a ≡ b mod n), and the relation “having the same image under a function” (f(a) = f(b)). Equivalence relations are in one-to-one correspondence with partitions of a set: each partition defines a unique equivalence relation, and each equivalence relation partitions the set into its equivalence classes.

Applications span algebra, topology, and computer science, where quotient structures and program or data equivalence are studied. Logical equivalence, where statements p and q satisfy p if and only if q, is another manifestation of the same formal idea.