Definition:Reflexive Closure
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Definition
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 $\RR$.
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.