Odd Integers under Addition do not form Group

Theorem
Let $S$ be the set of odd integers:
 * $S = \set {x \in \Z: \exists n \in \Z: x = 2 n + 1}$

Let $\struct {S, +}$ denote the algebraic structure formed by $S$ under the operation of addition.

Then $\struct {S, +}$ is not a group.

Proof
It is to be demonstrated that $\struct {S, +}$ does not satisfy the group axioms.

Let $a$ and $b$ be odd integers.

Then $a = 2 n + 1$ and $b = 2 m + 1$ for some $m, n \in \Z$.

Then:

and it is seen that $a + b$ is even.

That is:
 * $a + b \notin S$

Thus $\struct {S, +}$ does not fulfil.

Hence the result.