symmetric closure

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:

1. S is symmetric
2. R ⊆ S
3. 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}