Reflexive closure

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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[edit]

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

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