Group/Examples/x+y+2 over Reals

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Example of Group

Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:

$\forall x, y \in \R: x \circ y := x + y + 2$

Then $\struct {\R, \circ}$ is a group whose identity is $-2$.


Proof

Taking the group axioms in turn:

$G \, 0$: Closure

$\forall x, y \in \R: x + y + 2 \in \R$

Thus $x \circ y \in \R$ and so $\struct {\R, \circ}$ is closed.

$\Box$


$G \, 1$: Associativity

\(\displaystyle x \circ \paren {y \circ z}\) \(=\) \(\displaystyle x + \paren {y + z + 2} + 2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle x + y + z + 4\) $\quad$ $\quad$


\(\displaystyle \paren {x \circ y} \circ z\) \(=\) \(\displaystyle \paren {x + y + 2} + z + 2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle x + y + z + 4\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle x \circ \paren {y \circ z}\) $\quad$ $\quad$

Thus $\circ$ is associative.

$\Box$


$G \, 2$: Identity

Let $y$ be such that $x \circ y = x$.

Then:

\(\displaystyle x \circ y\) \(=\) \(\displaystyle x\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle x + y + 2\) \(=\) \(\displaystyle x\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(=\) \(\displaystyle -2\) $\quad$ $\quad$


Then it is noted that:

\(\displaystyle -2 \circ x\) \(=\) \(\displaystyle -2 + x + 2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle x\) $\quad$ $\quad$


Thus $-2$ is the identity element of $\struct {\R, \circ}$.

$\Box$


$G \, 3$: Inverses

We have that $-2$ is the identity element of $\struct {\R, \circ}$.

So:

\(\displaystyle x \circ y\) \(=\) \(\displaystyle -2\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle x + y + 2\) \(=\) \(\displaystyle -2\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(=\) \(\displaystyle -x - 4\) $\quad$ $\quad$


Then it is noted that:

\(\displaystyle \paren {-x - 4} \circ x\) \(=\) \(\displaystyle -x - 4 + x + 2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle -2\) $\quad$ $\quad$


Thus every element of $\struct {\R, \circ}$ has an inverse $-x - 4$.

$\Box$


All the group axioms are thus seen to be fulfilled, and so $\struct {\R, \circ}$ is a group.

$\blacksquare$


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