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


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


The reflexive closure of $\mathcal R$ is denoted $\mathcal R^=$.


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.