Category:Definitions/Reflexive Closures

From ProofWiki
Jump to navigation Jump to search

This category contains definitions related to Reflexive Closures.
Related results can be found in Category:Reflexive Closures.


Definition 1

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


The reflexive closure of $\RR$ is denoted $\RR^=$, and is defined as:

$\RR^= := \RR \cup \set {\tuple {x, x}: x \in S}$

That is:

$\RR^= := \RR \cup \Delta_S$

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

Let $\QQ$ be the set of all reflexive relations on $S$ that contain $\RR$.

The reflexive closure of $\RR$ is denoted $\RR^=$, and is defined as:

$\RR^= := \bigcap \QQ$

That is:

$\RR^=$ is the intersection of all reflexive relations on $S$ containing $\RR$.