Group/Examples/x+y+2 over Reals
Jump to navigation
Jump to search
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:
Group Axiom $\text G 0$: Closure
$\forall x, y \in \R: x + y + 2 \in \R$
Thus $x \circ y \in \R$ and so $\struct {\R, \circ}$ is closed.
$\Box$
Group Axiom $\text G 1$: Associativity
\(\ds x \circ \paren {y \circ z}\) | \(=\) | \(\ds x + \paren {y + z + 2} + 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x + y + z + 4\) |
\(\ds \paren {x \circ y} \circ z\) | \(=\) | \(\ds \paren {x + y + 2} + z + 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x + y + z + 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \circ \paren {y \circ z}\) |
Thus $\circ$ is associative.
$\Box$
Group Axiom $\text G 2$: Existence of Identity Element
Let $y$ be such that $x \circ y = x$.
Then:
\(\ds x \circ y\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x + y + 2\) | \(=\) | \(\ds x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds -2\) |
Then it is noted that:
\(\ds -2 \circ x\) | \(=\) | \(\ds -2 + x + 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x\) |
Thus $-2$ is the identity element of $\struct {\R, \circ}$.
$\Box$
Group Axiom $\text G 3$: Existence of Inverse Element
We have that $-2$ is the identity element of $\struct {\R, \circ}$.
So:
\(\ds x \circ y\) | \(=\) | \(\ds -2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x + y + 2\) | \(=\) | \(\ds -2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds -x - 4\) |
Then it is noted that:
\(\ds \paren {-x - 4} \circ x\) | \(=\) | \(\ds -x - 4 + x + 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -2\) |
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.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Exercise $1 \ \text{(b)}$