Example:Antiassociative Structure

Theorem
Let $\left({\R_{\ne 0}, \circ}\right)$ be an algebraic structure where $\R_{\ne 0}$ denotes the non-zero real numbers.

Define:


 * $\forall x, y \in \R_{\ne 0}: x \circ y = xy + y$

Then $\circ$ is antiassociative on $\R_{\ne 0}$, that is:


 * $\forall x, y, z \in \R_{\ne 0}: \left({x \circ y}\right) \circ z = x \circ \left({y \circ z}\right)$

Proof
Let $a, b, c \in \R_{\ne 0}$:

Then:

And:

As $ac \ne 0$


 * $\forall x, y, z \in \R_{\ne 0}: \left({x \circ y}\right) \circ z = x \circ \left({y \circ z}\right)$

Hence the result.