nonreflexive

Nonreflexive is a term used in logic and mathematics to describe a binary relation that is not reflexive. If a relation R is defined on a set A, it is reflexive when every element relates to itself: for all x in A, xRx holds. If there exists at least one element a in A such that not (aRa), the relation is nonreflexive. In other words, nonreflexive means the relation is not reflexive; it does not require that no element relates to itself, only that some element fails to relate to itself.

Nonreflexive relations can still include some self-relations. For example, on A = {1, 2}, the relation R = { (1,1) } is nonreflexive because the pair (2,2) is missing, but it is not irreflexive since (1,1) is in R. By contrast, a relation that is irreflexive requires that no element relates to itself: for all x in A, not (xRx). Irreflexive relations are a stricter form of nonreflexive.

Common examples help illustrate the distinction. The standard “greater than” relation on numbers is irreflexive and hence nonreflexive, since no number is greater than itself. The empty relation on any set is irreflexive, and therefore nonreflexive as well. Conversely, the usual “less than or equal to” relation is reflexive, so it is not nonreflexive.

Understanding nonreflexive properties aids in the study of relation categories, including reflexive, irreflexive, and other combinations, and informs proofs in set theory, graph theory, and order theory.