The symmetric closure of $\mathord {\sim }_{R}$ is the relation $\smash {\mathord {\sim }^{\mathrm{symm}}_{R}}$1 satisfying the following universal property:2
- Given another symmetric relation $\mathord {\sim }_{S}$ on $A$ such that $R\subset S$, there exists an inclusion $\smash {\mathord {\sim }^{\mathrm{symm}}_{R}}\subset \mathord {\sim }_{S}$.
1Further Notation: Also written $R^{\mathrm{symm}}$.
2Slogan: The symmetric closure of $R$ is the smallest symmetric relation containing $R$.
