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 $$R.$$ A relation is called if it relates every element of $$X$$ to itself.

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

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

Example
As an example, if $$X = \{1, 2, 3, 4\}$$ $$R = \{(1,1), (2,2), (3,3), (4,4)\}$$ then the relation $$R$$ is already reflexive by itself, so it does not 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 on the same set $$X$$ $$R = \{(1,1), (2,2), (4,4)\}$$ then the reflexive closure is $$S = R \cup \{(x,x): x \in X\} = \{(1,1), (2,2), (3,3), (4,4)\} .$$