Associative and anticommutative Binary Operation(composition)

Question

Show that if binary operation ,$\Delta$, is associative and anticommutative on $\mathbb{E}$, then $x\Delta y \Delta z=x\Delta z$ ∀$x,y,z \in \mathbb{E}$. [Hint: consider $x\Delta y\Delta z\Delta x\Delta z$]

Answer 1

Since $\Delta$ is associative and anticommutative we have $\Delta$ is idempotent and $x \Delta y \Delta x=x$ ∀$x,y \in \mathbb{E}$ see Associative Binary Operation(composition) is anticommutative iff idempotent...

∴ we right multiply $x \Delta y \Delta x=x$ by $y,z,x,z$ in that order.

This yields $x \Delta y \Delta x\Delta y\Delta z\Delta x\Delta z=x\Delta y\Delta z\Delta x\Delta z$ $\Longrightarrow$ $(x \Delta y \Delta x)\Delta y(\Delta z\Delta x\Delta z)=(x\Delta( y\Delta z)\Delta x)\Delta z$ $\Longrightarrow$ $x\Delta y \Delta z=x\Delta z$

QED