nLab
symmetric relation

Contents

Definition

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

(x,y:A),xyyx\forall (x, y: A),\; x \sim y \;\Rightarrow\; y \sim x

In the language of the 22-poset-with-duals Rel of sets and relations, a relation R:AAR: A \to A is symmetric if it is contained in its reverse:

RR opR \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)