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

Example

As an example, if

$X = \left\{ 1, 2, 3, 4 \right\}$

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

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

$X = \left\{ 1, 2, 3, 4 \right\}$

$R = \left\{ (1,1), (2,2), (4,4) \right\}$

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

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

See also

• Transitive closure
• Symmetric closure

References

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