Bijection/Examples/Negative Functions

Example of Bijection

Let $\mathbb S$ be one of the standard number systems $\Z$, $\Q$, $\R$, $\C$.

Let $h: \mathbb S \to \mathbb S$ be the negation function defined on $\mathbb S$:

$\forall x \in \mathbb S: \map h x = -x$

Then $h$ is a bijection.

Proof

Complex Numbers

By the definition of the negative of complex number, the complex negation function is defined on the complex numbers $\C$ as:

$-z := -x - i y$

Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2 \in \C$ such that $\map h {z_1} = \map h {z_2}$.

\(\ds \map h {z_1}\)

\(=\)

\(\ds \map h {z_2}\)

\(\ds \leadsto \ \ \)

\(\ds -z_1\)

\(=\)

\(\ds -z_2\)

\(\ds \leadsto \ \ \)

\(\ds -x_1 - i y_1\)

\(=\)

\(\ds -x_2 - i y_2\)

\(\ds \leadsto \ \ \)

\(\ds -x_1\)

\(=\)

\(\ds -x_2\)

equating real parts

\(\, \ds \land \, \)

\(\ds -y_1\)

\(=\)

\(\ds -y_2\)

equating imaginary parts

\(\ds \leadsto \ \ \)

\(\ds x_1\)

\(=\)

\(\ds x_2\)

Real Negation Function is Bijection

\(\, \ds \land \, \)

\(\ds y_1\)

\(=\)

\(\ds y_2\)

This needs considerable tedious hard slog to complete it. In particular: too tedious for my current level of attention span To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions