In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation S on X that contains R. Restated:

- S is symmetric
- R ⊆ S
- For any relation S’, if R ⊆ S’ and S’ is symmetric then S ⊆ S’. In other words, S is smallest relation that satisfies 1 and 2.

The symmetric closure S of a binary relation R on a set X can be formally defined as:

S = R ∪ {(*x*, *y*) : (*y*, *x*) ∈ R}

Where {(*x*, *y*) : (*y*, *x*) ∈ R} is the inverse relation of R, R^{-1}. The symmetric closure of a binary relation on a set is the union of the binary relation and it’s inverse.

For example, let R be the greater than relation on the set of integers I:

R = {(*a*, *b*) | *a* ∈ I ∧ *b* ∈ I ∧ *a* > *b*}

The symmetric closure is the union of the relation (greater than) and its inverse (less than) over I. The union of greater than and less than is not equal to, ≥ or ≠. So the symmetric closure S of the greater than relation R is:

S = {(*a*, *b*) | *a* ∈ I ∧ *b* ∈ I ∧ *a <> **b*} = {(*a*, *b*) | *a* ∈ I ∧ *b* ∈ I ∧ *a ≠ **b*}