# Notation for symmetric closure of a relation

Given a binary relation $$r$$, its reflexive closure, $$r \cup id$$, is sometimes written as $$r^?$$ or $$r^=$$. Its transitive closure is written as $$r^+$$. Its reflexive, transitive closure is written as $$r^*$$.

What about its symmetric closure, $$r \cup r^{-1}$$? Is there any existing notation for that? I saw $$s(r)$$ once, but that's not particularly appealing to me. Are there any other candidates?

• I've never seen one. I don't visit places where I would be likely to, so haven't seen the others either. If you need the concept often in something you are writing, invent notation and tell your reader. May 29, 2019 at 12:38

I would use $$R^s$$ for the symmetrisation of a relation $$R$$, as in this link.
By symmetrization of the binary relation $$R$$ on a set one obtains the symmetric relation $$R^s$$ defined by: $$a \mathrel{R^s} b$$ if and only if $$a \mathrel{R} b$$ or $$b \mathrel{R} a$$, such that every symmetric relation implying $$\mathrel{R}$$ also implies $$\mathrel{R^s}$$.