Nietequivalence

Nietequivalence is a term sometimes used in mathematical and theoretical computer science literature to refer to a binary relation that is not an equivalence relation. The prefix niet- signals “not” or a deviation from the standard notion of equivalence. Because it is not a standardized term, its precise meaning can vary by author; in general it denotes a relation that fails one or more of the three axioms of an equivalence relation: reflexivity, symmetry, and transitivity.

A typical way to describe a nietequivalence is to specify a relation that is not fully reflexive, symmetric, or transitive. For example, a relation may be reflexive but not symmetric, or symmetric but not transitive, or fail all three properties. The lack of one or more axioms distinguishes het from a bona fide equivalence relation and from other non-equivalence notions such as partial orders.

Example: on the set of integers Z, define R by xRy if x − y ∈ {0, 1}. This relation is reflexive since x − x = 0 ∈ {0,1}. It is not symmetric because if x − y = 1 then y − x = −1 ∉ {0,1}. It is not transitive since x − y = 1 and y − z = 1 would give x − z = 2 ∉ {0,1}.

Usage and context: nietequivalence is used to discuss relationships that resemble equality but do not satisfy all equivalence axioms. It can model approximate equality in numerical computations, tolerance relations in data analysis, or context-dependent notions of sameness in programming languages. Because the term is informal and varyingly defined, readers should check the author's specific definition when encountering it.