Monoid/Examples/x+y+xy on Reals
Jump to navigation
Jump to search
Example of Monoid
Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:
- $\forall x, y \in \R: x \circ y := x + y + x y$
Then $\struct {\R, \circ}$ is a monoid whose identity is $0$.
Proof
We have that:
- $\forall x, y \in \R: x \circ y \in \R$
and so $\struct {\R, \circ}$ is closed.
Now let $x, y, z \in \R$.
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 $\circ$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x + \paren {y + z + y z} + \paren {x y + x z + x y z}\) | Real Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds x + y + z + y z + x y + x z + x y z\) | Real Addition is Associative |
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 $\circ$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x + y + x y} + z + \paren {x z + y z + x y z}\) | Real Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds x + y + x y + z + x z + y z + x y z\) | Real Addition is Associative |
As can be seen by inspection:
- $x \circ \paren {y \circ z} = \paren {x \circ y} \circ z$
and so $\circ$ is associative.
Then we have:
\(\ds x \circ 0\) | \(=\) | \(\ds x + 0 + x \times 0\) | Definition of $\circ$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x\) | Real Addition Identity is Zero, Zero Element of Multiplication on Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 + x + 0 \times x\) | Real Addition Identity is Zero, Zero Element of Multiplication on Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 \circ x\) | Definition of $\circ$ |
The result follows by definition of monoid.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $4$. Groups: Exercise $11$