In mathematics, a binary relation ${\displaystyle R}$ on a set ${\displaystyle X}$ is reflexive if it relates every element of ${\displaystyle X}$ to itself.

An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.

The word reflexive is originally derived from the Medieval Latin reflexivus ('recoiling' [cf. reflex], or 'directed upon itself') (c. 1250 AD) from the classical Latin reflexus- ('turn away', 'reflection') + -īvus (suffix). The word entered Early Modern English in the 1580s. The sense of the word meaning 'directed upon itself', as now used in mathematics, surviving mostly by its use in philosophy and grammar (cf. Reflexive verb and Reflexive pronoun).

The first explicit use of "reflexivity", that is, describing a relation as having the property that every element is related to itself, is generally attributed to Giuseppe Peano in his Arithmetices principia (1889), wherein he defines one of the fundamental properties of equality being ${\displaystyle a=a}$. The first use of the word reflexive in the sense of mathematics and logic was by Bertrand Russell in his Principles of Mathematics (1903).

A relation ${\displaystyle R}$ on the set ${\displaystyle X}$ is said to be reflexive if for every ${\displaystyle x\in X}$, ${\displaystyle (x,x)\in R}$.

Equivalently, letting ${\displaystyle \operatorname {I} _{X}:=\{(x,x)~:~x\in X\}}$ denote the identity relation on ${\displaystyle X}$, the relation ${\displaystyle R}$ is reflexive if ${\displaystyle \operatorname {I} _{X}\subseteq R}$.

The reflexive closure of ${\displaystyle R}$ is the union ${\displaystyle R\cup \operatorname {I} _{X},}$ which can equivalently be defined as the smallest (with respect to ${\displaystyle \subseteq }$) reflexive relation on ${\displaystyle X}$ that is a superset of ${\displaystyle R.}$ A relation ${\displaystyle R}$ is reflexive if and only if it is equal to its reflexive closure.

The reflexive reduction or irreflexive kernel of ${\displaystyle R}$ is the smallest (with respect to ${\displaystyle \subseteq }$) relation on ${\displaystyle X}$ that has the same reflexive closure as ${\displaystyle R.}$ It is equal to ${\displaystyle R\setminus \operatorname {I} _{X}=\{(x,y)\in R~:~x\neq y\}.}$ The reflexive reduction of ${\displaystyle R}$ can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of ${\displaystyle R.}$ For example, the reflexive closure of the canonical strict inequality ${\displaystyle <}$ on the reals ${\displaystyle \mathbb {R} }$ is the usual non-strict inequality ${\displaystyle \leq }$ whereas the reflexive reduction of ${\displaystyle \leq }$ is ${\displaystyle <.}$