Associative Operation/Examples/Non-Associative/xy+1

Example of Non-Associative Operations
Let $\R$ denote the set of real numbers.

Let $\circ$ denote the operation on $\R$ defined as:
 * $\forall x, y \in \R: x \circ y := x y + 1$

Then $\circ$ is not an associative operation, despite being commutative.

Proof
Let $x, y, z \in \R$.

We have:

But unless $x = z$ it is not the case that $\paren {x y + 1} z + 1 = x y z + x + 1$.

However, $x y + 1 = y x + 1$ and so $\circ$ is commutative.