Group/Examples/x+y+2 over Reals

Example of Group
Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:
 * $\forall x, y \in \R: x \circ y := x + y + 2$

Then $\struct {\R, \circ}$ is a group whose identity is $-2$.

Proof
Taking the group axioms in turn:

$G \, 0$: Closure
$\forall x, y \in \R: x + y + 2 \in \R$

Thus $x \circ y \in \R$ and so $\struct {\R, \circ}$ is closed.

$G \, 1$: Associativity
Thus $\circ$ is associative.

$G \, 2$: Identity
Let $y$ be such that $x \circ y = x$.

Then:

Then it is noted that:

Thus $-2$ is the identity element of $\struct {\R, \circ}$.

$G \, 3$: Inverses
We have that $-2$ is the identity element of $\struct {\R, \circ}$.

So:

Then it is noted that:

Thus every element of $\struct {\R, \circ}$ has an inverse $-x - 4$.

All the group axioms are thus seen to be fulfilled, and so $\struct {\R, \circ}$ is a group.