Semigroup/Examples/x+y-xy on Integers

Example of Semigroup
Let $\circ: \Z \times \Z$ be the operation defined on the integers $\Z$ as:
 * $\forall x, y \in \Z: x \circ y := x + y - x y$

Then $\struct {\Z, \circ}$ is a semigroup.

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

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

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

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.

The result follows by definition of semigroup.

Also see

 * Inclusion-Exclusion Principle (think about why)