# Anticommutative property

In mathematics, **anticommutativity** is a specific property of some non-commutative operations. In mathematical physics, where symmetry is of central importance, these operations are mostly called **antisymmetric operations**, and are extended in an associative setting to cover more than two arguments. Swapping the position of two arguments of an antisymmetric operation yields a result which is the *inverse* of the result with unswapped arguments. The notion *inverse* refers to a group structure on the operation's codomain, possibly with another operation, such as addition.

Subtraction is an anticommutative operation because −(*a* − *b*) = *b* − *a*. For example, 2 − 10 = −(10 − 2) = −8.

A prominent example of an anticommutative operation is the Lie bracket.

and the proof in the multilinear case is the same but in only two of the inputs.