# Definition:Reflexive Closure

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## Definition

### Definition 1

Let $\mathcal R$ be a relation on a set $S$.

The **reflexive closure** of $\mathcal R$ is denoted $\mathcal R^=$, and is defined as:

- $\mathcal R^= := \mathcal R \cup \set {\tuple {x, x}: x \in S}$

That is:

- $\mathcal R^= := \mathcal R \cup \Delta_S$

where $\Delta_S$ is the diagonal relation on $S$.

### Definition 2

Let $\RR$ be a relation on a set $S$.

The **reflexive closure** of $\RR$ is defined as the smallest reflexive relation on $S$ that contains $\RR$ as a subset.

The **reflexive closure** of $\RR$ is denoted $\RR^=$.

### Definition 3

Let $\mathcal R$ be a relation on a set $S$.

Let $\mathcal Q$ be the set of all reflexive relations on $S$ that contain $\mathcal R$.

The **reflexive closure** of $\mathcal R$ is denoted $\mathcal R^=$, and is defined as:

- $\mathcal R^= := \bigcap \mathcal Q$

That is:

- $\mathcal R^=$ is the intersection of all reflexive relations on $S$ containing $\mathcal R$.

## Equivalence of Definitions

The above definitions are all equivalent, as shown on Equivalence of Definitions of Reflexive Closure.

## Also see

- Results about
**reflexive closures**can be found here.