Monoid/Examples/x+y+xy on Reals

Example of Monoid
Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:
 * $\forall x, y \in \R: x \circ y := x + y + x y$

Then $\struct {\R, \circ}$ is a monoid whose identity is $0$.

Proof
We have that:
 * $\forall x, y \in \R: x \circ y \in \R$

and so $\struct {\R, \circ}$ is closed.

Now let $x, y, z \in \R$.

We have:

and:

As can be seen by inspection:
 * $x \circ \paren {y \circ z} = \paren {x \circ y} \circ z$

and so $\circ$ is associative.

Then we have:

The result follows by definition of monoid.