A relation $R$ on $A$ is symmetric if we have $R^{\dagger }=R$.
In detail, a relation $R$ is symmetric if it satisfies the following condition:
- (★) For each $a,b\in A$, if $a\sim _{R}b$, then $b\sim _{R}a$.
Let $A$ be a set.
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1.
The set of symmetric relations on $A$ is the subset $\smash {\mathrm{Rel}^{\mathrm{symm}}\webleft (A,A\webright )}$ of $\mathrm{Rel}\webleft (A,A\webright )$ spanned by the symmetric relations.
-
2.
The poset of relations on $A$ is is the subposet $\smash {\mathbf{Rel}^{\mathsf{symm}}\webleft (A,A\webright )}$ of $\mathbf{Rel}\webleft (A,A\webright )$ spanned by the symmetric relations.
Let $R$ and $S$ be relations on $A$.
Proof of Proposition 10.3.1.1.4.
Item 1: Interaction With Inverses
Clear.
Item 2: Interaction With Composition
Clear.
