# Associative and Anticommutative

## Theorem

Let $\circ$ be a binary operation on a set $S$.

Let $\circ$ be both associative and anticommutative.

Then:

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

## Proof

Let $\circ$ be both associative and anticommutative.

Then from Associative Idempotent Anticommutative:

$\forall x, z \in S: x \circ z \circ x = x$

and $\circ$ is idempotent.

Consider $x \circ y \circ z \circ x \circ z$.

We have:

 $\displaystyle x \circ y \circ z \circ x \circ z$ $=$ $\displaystyle x \circ \left({y \circ z}\right) \circ x \circ z$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle x \circ z$ $\quad$ $\quad$

Also:

 $\displaystyle x \circ y \circ z \circ x \circ z$ $=$ $\displaystyle x \circ y \circ \left({z \circ x \circ z}\right)$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle x \circ y \circ z$ $\quad$ $\quad$

Hence the result.

$\blacksquare$