Example:Antiassociative Structure

Theorem
Let $\left({\R_{>0}, \circ}\right)$ be an algebraic structure where $\R_{>0}$ denotes the strictly positive real numbers.

Define:


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

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


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

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

Then:

and:

As $ac \ne 0$:


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

Hence the result.