Reflexive Reduction of Relation Compatible with Group Operation is Compatible

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {S, \circ}$ be a group.

Let $\RR$ be a relation on $S$ which is compatible with $\circ$.

Let $\RR^\ne$ be the reflexive reduction of $\RR$.


Then $\RR^\ne$ is compatible with $\circ$.


Proof

By definition of reflexive reduction, for all $a, b \in S$:

$a \mathrel {\RR^\ne} b$ if and only if $a \mathrel \RR b$ but $a \ne b$.

By definition of the diagonal relation $\Delta_S$:

$a \ne b$ if and only if $\tuple {a, b} \notin \Delta_S$

Thus, considered as subsets of $S \times S$, we have:

$\RR^\ne = \RR \setminus \Delta_S$


By Diagonal Relation is Universally Compatible, $\Delta_S$ is compatible with $\circ$.


Thus by Set Difference of Relations Compatible with Group Operation is Compatible, $\RR^\ne$ is compatible with $\circ$.

$\blacksquare$