# Contents

## Definition

A (binary) relation $\sim$ on a set $A$ is symmetric if any two elements that are related in one order are also related in the other order:

$\forall \left(x,y:A\right),\phantom{\rule{thickmathspace}{0ex}}x\sim y\phantom{\rule{thickmathspace}{0ex}}⇒\phantom{\rule{thickmathspace}{0ex}}y\sim x$\forall (x, y: A),\; x \sim y \;\Rightarrow\; y \sim x

In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R:A\to A$ is symmetric if it is contained in its reverse:

$R\subseteq {R}^{\mathrm{op}}$R \subseteq R^{op}

In that case, this containment is in fact an equality.

Revised on August 24, 2012 20:04:06 by Urs Schreiber (89.204.138.8)