Semigroup/Examples/x+y-xy on Integers
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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:
\(\ds x \circ \paren {y \circ z}\) | \(=\) | \(\ds x + \paren {y \circ z} - x \paren {y \circ z}\) | Definition of $\circ$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x + \paren {y + z - y z} - x \paren {y + z - y z}\) | Definition of $\circ$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x + y + z - y z - x y - x z + x y z\) |
and:
\(\ds \paren {x \circ y} \circ z\) | \(=\) | \(\ds \paren {x \circ y} + z - \paren {x \circ y} z\) | Definition of $\circ$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x + y - x y} + z - \paren {x + y - x y} z\) | Definition of $\circ$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x + y - x y + z - x z - y z + x y z\) |
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.
$\blacksquare$
Also see
- Inclusion-Exclusion Principle (think about why)
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Semigroups: Exercise $1$