Definition:Anticommutative

Let $$\circ$$ be a binary operation.

Then $$\circ$$ is anticommutative on $$S$$ iff

$$\forall x, y \in S: x \circ y = y \circ x \Longrightarrow x = y$$

Equivalently, it can be defined as:

$$\forall x, y \in S: x \ne y \Longrightarrow x \circ y \ne y \circ x$$

An example of an anticommutative operation in arithmetic is the operation of subtraction.