Associative Binary Operation from associative Binary Operation

Question

if $\Delta$ is an associative composition(Binary Operation) on $\mathbb{E}$ and if $a\in \mathbb{E}$, then the composition $\Omega$ on $\mathbb{E}$ defined by $x\Omega y=x\Delta a\Delta y$ is associative.

Answer 1

We want to show if $x,y,z\in \mathbb{E}$, then $(x\Omega y)\Omega z=x\Omega(y\Omega z)$ .

$(x\Omega y)\Omega z=(x\Omega y)\Delta a\Delta z=x\Delta a \Delta y \Delta a \Delta z=x\Delta a \Delta (y \Delta a \Delta z)=x\Delta a \Delta (y\Omega z)=x\Omega(y\Omega z) $.