A relation $R$ on $A$ is symmetric if we have $R^{\dagger }=R$.
- (★) For each $a,b\in A$, if $a\sim _{R}b$, then $b\sim _{R}a$.
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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.
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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.
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1.
Interaction With Inverses. If $R$ is symmetric, then so is $R^{\dagger }$.
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2.
Interaction With Composition. If $R$ and $S$ are symmetric, then so is $S\mathbin {\diamond }R$.
10.3.1 Foundations
Let $A$ be a set.
In detail, a relation $R$ is symmetric if it satisfies the following condition:
Let $A$ be a set.
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.
