Monoid/Examples/x+y+xy on Reals

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 $11$