show that $\nabla $ is commutative Let $ \mathcal S$ be a non empty set and $ \nabla$ a binary algebraic action such as $\forall x,y \in \mathcal S$ the following conditions are true: 
  $ x \nabla(x \nabla y)=y  $ and $ (y \nabla x)\nabla x=y$
Show that $\nabla$ is commutative but not necessary associative.
I know I am supposed to show some work but I have no idea what to do.
Please , could anybody help?
 A: Proof of Commutativity
Let $x,y\in\mathcal{S}$.  From the second identity, $\big(y\nabla(y\nabla x)\big)\nabla (y\nabla x)=y$.  Hence, by the first identity, $x\nabla (y\nabla x)=y$.  

 Therefore, $x\nabla \big(x\nabla (y\nabla x)\big)=x\nabla y$.  However, the first identity demands that $x\nabla \big(x\nabla (y\nabla x)\big)=y\nabla x$.


Counterexample for Associativity
Take $\mathcal{S}:=\mathbb{R}$ and assume that there are constants $a,b\in\mathbb{R}$ such that $x\nabla y:=ax+by$ for all $x,y\in\mathcal{S}$. Find such $a$ and $b$ that satisfy the two required identities.

  It is easily shown that $a=-1$ and $b=-1$ work.  Then, for $x,y,z\in\mathcal{S}$, we have $(x\nabla y)\nabla z=x\nabla (y\nabla z)$ if and only if $x=z$.

A: Brute force is common for problems like these.
Since you only have one operation, I'm going to write it as juxtaposition; e.g. your first identity is $x(xy) = y$.
Write down a few elements of the simplest expressions in the structure (e.g. $x, y, z, xy, xx, \ldots$).
Plug in a few of the simplest ones in all combinations into the identities to see if there are any simplifications you can do and new identities you can derive.
Similarly, in various equations, try to make substitutions to create situations you can apply other identities to. e.g. in this example if you have an equation that has $(xy)$ somewhere in it, you might try substituting $y \mapsto (xy)$ so as to apply your first identity in the forward direction.
Often, if you can write down something that simplifies in two different ways, you produce a brand new identity.
(to prove it's not associative, come up with a model)
