# Group/Examples/inv x = 1 - x

## Contents

## Theorem

Let $S = \set {x \in \R: 0 < x < 1}$.

Then an operation $\circ$ can be found such that $\struct {S, \circ}$ is a group such that the inverse of $x \in S$ is $1 - x$.

## Proof

Define $f: \openint 0 1 \to \R$ by:

- $\map f x := \map \ln {\dfrac {1 - x} x}$

Let us show that $f$ is a bijection by constructing an inverse mapping $g: \R \to \openint 0 1$:

- $\map g z := \dfrac 1 {1 + \exp z}$

### Lemma 1

- $\map {f \circ g} x = x$

### Lemma 2

- $\map {g \circ f} x = x$

Thus $f$ is a bijection.

Let $\struct {\R, +}$ be the additive group on $\R$.

Now define $\circ := +_f$ to be the operation induced on $\openint 0 1$ by $f$ and $+$:

- $x \circ y := \map {f^{-1} } {\map f x + \map f y}$

Let us determine the behaviour of $\circ$ more explicitly:

\(\displaystyle x \circ y\) | \(=\) | \(\displaystyle \map g {\map f x + \map f y}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {1 + \map \exp {\map \log {\frac {1 - x} x} + \map \log {\frac {1 - y} y} } }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {1 + \map \exp {\map \log {\frac {1 - x} x} } \map \exp {\map \log {\frac {1 - y} y} } }\) | Exponential of Sum | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {1 + \paren {\frac {1 - x} x} \paren {\frac {1 - y} y} }\) | Exponential of Natural Logarithm |

We see that $\circ$ is commutative.

Let $x = \dfrac 1 2$, $\dfrac {1 - x} x = 1$ so that:

\(\displaystyle \dfrac 1 2 \circ y\) | \(=\) | \(\displaystyle \dfrac 1 {1 + \paren {\frac {1 - y} y} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {\frac y y + \frac {1 - y} y}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {\frac 1 y}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle y\) |

so that $\dfrac 1 2$ is the identity element for $\circ$.

Furthermore, putting $y = 1 - x$, the following obtains:

\(\displaystyle \frac 1 {1 + \paren {\frac {1 - x} x} \paren {\frac x {1 - x} } }\) | \(=\) | \(\displaystyle \frac 1 {1 + 1}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 2\) |

establishing $1 - x$ to be the inverse of $x$, as desired.

That $\circ$ in fact determines a group on $\openint 0 1$ follows from Pullback of Group is Group.

$\blacksquare$

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26 \mu$