nLab symmetric relation




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.

Relation to graphs

A set with a symmetric relation is the same as a loop digraph (V,E,s:EV,t:EV)(V, E, s:E \to V, t:E \to V) with a function sym:EEsym:E \to E such that

  • for every fEf \in E, s(f)= Vt(sym(f))s(f) =_V t(sym(f))
  • for every fEf \in E, t(f)= Vs(sym(f))t(f) =_V s(sym(f))
  • for every fEf \in E, sym(sym(f))= Efsym(sym(f)) =_E f

Last revised on September 22, 2022 at 18:42:01. See the history of this page for a list of all contributions to it.