Reflexive closure

In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X that contains R.

For example, if X is a set of distinct numbers and x R y means "x is less than y", then the reflexive closure of R is the relation "x is less than or equal to y".

Definition

The reflexive closure S of a relation R on a set X is given by

\( S=R\cup \left\{(x,x):x\in X\right\} \)

In English, the reflexive closure of R is the union of R with the identity relation on X.

Example

As an example, if

\( {\displaystyle X=\left\{1,2,3,4\right\}} \)
\( {\displaystyle R=\left\{(1,1),(2,2),(3,3),(4,4)\right\}} \)

then the relation R is already reflexive by itself, so it doesn't differ from its reflexive closure.

However, if any of the pairs in R was absent, it would be inserted for the reflexive closure. For example, if

\( {\displaystyle X=\left\{1,2,3,4\right\}} \)
\( {\displaystyle R=\left\{(1,1),(2,2),(4,4)\right\}} \)

then reflexive closure is, by the definition of a reflexive closure:

\( {\displaystyle S=R\cup \left\{(x,x):x\in X\right\}=\left\{(1,1),(2,2),(3,3),(4,4)\right\}}. \)