Reflexive relation

Lore & Background

The word 'reflexive' derives from Medieval Latin reflexivus ('recoiling' or 'directed upon itself') from around 1250 AD, from classical Latin reflexus- ('turn away', 'reflection') plus the suffix -īvus. It entered Early Modern English in the 1580s, with the sense of 'directed upon itself' surviving in mathematics, philosophy, and grammar. The first explicit use of 'reflexivity' to describe a relation where every element is related to itself is generally attributed to Giuseppe Peano in his Arithmetices principia (1889), where he defines one of the fundamental properties of equality as a = a. The first use of the word 'reflexive' in the sense of mathematics and logic was by Bertrand Russell in his Principles of Mathematics (1903).

Reader's Guide

The concept of a reflexive relation is foundational in mathematics, particularly in set theory and the study of equivalence relations. A relation R on a set X is reflexive if for every x in X, the pair (x,x) is in R. This is equivalent to the identity relation on X being a subset of R. The reflexive closure of a relation R is the smallest reflexive relation containing R, formed by taking the union of R with the identity relation. Conversely, the reflexive reduction (or irreflexive kernel) removes all pairs (x,x) from R. Related definitions include irreflexive (no element relates to itself), quasi-reflexive (if xRy then xRx and yRy), and coreflexive (if xRy then x=y). A reflexive relation on a nonempty set cannot be irreflexive, asymmetric, or antitransitive. These distinctions are crucial for classifying relations and understanding their structural properties in algebra, logic, and computer science.

Did You Know?

• The reflexive closure of the strict inequality < on the reals is the non-strict inequality ≤.
• A relation is irreflexive if and only if its complement in X × X is reflexive.
• An asymmetric relation is necessarily irreflexive.
• A transitive and irreflexive relation is necessarily asymmetric.