Prove that certain operation is commutative

Question

Suppose we have a set $X$ and a binary operation $\circ : X \times X \to X$, such that $\forall x,y \in X$ the following equalities hold $$y \circ (y\circ x) = x, (x\circ y)\circ y = x.$$

How can prove that this operation is commutative?

Answer 1

Goal: $x \circ y = y \circ x$.

Let $x \circ y = z$. Notice that $(x \circ y) \circ y = z \circ y = x$. Also notice that $z \circ (z \circ y) = z \circ x = y$. Finally we calculate $y \circ x = (z \circ x) \circ x = z = x \circ y$.