# 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.

$\Box$

### $G \, 1$: Associativity

 $\displaystyle x \circ \paren {y \circ z}$ $=$ $\displaystyle x + \paren {y + z + 2} + 2$ $\displaystyle$ $=$ $\displaystyle x + y + z + 4$

 $\displaystyle \paren {x \circ y} \circ z$ $=$ $\displaystyle \paren {x + y + 2} + z + 2$ $\displaystyle$ $=$ $\displaystyle x + y + z + 4$ $\displaystyle$ $=$ $\displaystyle x \circ \paren {y \circ z}$

Thus $\circ$ is associative.

$\Box$

### $G \, 2$: Identity

Let $y$ be such that $x \circ y = x$.

Then:

 $\displaystyle x \circ y$ $=$ $\displaystyle x$ $\displaystyle \leadsto \ \$ $\displaystyle x + y + 2$ $=$ $\displaystyle x$ $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle -2$

Then it is noted that:

 $\displaystyle -2 \circ x$ $=$ $\displaystyle -2 + x + 2$ $\displaystyle$ $=$ $\displaystyle x$

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

$\Box$

### $G \, 3$: Inverses

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

So:

 $\displaystyle x \circ y$ $=$ $\displaystyle -2$ $\displaystyle \leadsto \ \$ $\displaystyle x + y + 2$ $=$ $\displaystyle -2$ $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle -x - 4$

Then it is noted that:

 $\displaystyle \paren {-x - 4} \circ x$ $=$ $\displaystyle -x - 4 + x + 2$ $\displaystyle$ $=$ $\displaystyle -2$

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

$\Box$

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

$\blacksquare$