# 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

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

## Example

As an example, if

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

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

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

- .

## See also

## References

- Franz Baader and Tobias Nipkow,
*Term Rewriting and All That*, Cambridge University Press, 1998, p. 8

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