Reflexive Closure is Idempotent

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $S$ be a set.

Let $R$ denote the set of all endorelations on $S$.

Then the reflexive closure operator is an idempotent mapping on $R$.

That is:

$\forall \RR \in R: \RR^= = \paren {\RR^=}^=$

where $\RR^=$ denotes the reflexive closure of $\RR$.


Proof

Let $\RR \in R$.

By the definition of reflexive closure:

$\RR^= = \RR \cup \Delta_S$
$\paren {\RR^=}^= = \paren {\RR \cup \Delta_S} \cup \Delta_S$

By Union is Associative:

$\paren {\RR^=}^= = \RR \cup \paren {\Delta_S \cup \Delta_S}$

By Set Union is Idempotent:

$\paren {\RR^=}^= = \RR \cup \Delta_S$

Hence:

$\forall \RR \in R: \RR^= = \paren {\RR^=}^=$

$\blacksquare$