Construction 8.2.2.1.2 (01S0): The Symmetric Closure of a Relation

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Table of Contents
Part 2: Relations
Chapter 8: Equivalence Relations and Apartness Relations
Section 8.2: Symmetric Relations
Subsection 8.2.2: The Symmetric Closure of a Relation
Construction 8.2.2.1.2: The Symmetric Closure of a Relation (cite)

Construction 8.2.2.1.2 ▶ The Symmetric Closure of a Relation

Concretely, $\smash {\mathord {\sim }^{\mathrm{symm}}_{R}}$ is the symmetric relation on $A$ defined by

\begin{align*} R^{\mathrm{symm}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R\cup R^{\dagger }\\ & = \webleft\{ \webleft (a,b\webright )\in A\times A\ \middle |\ \text{we have $a\sim _{R}b$ or $b\sim _{R}a$}\webright\} .\end{align*}

Proof of Construction 8.2.2.1.2.

Clear.

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