# 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 $\, \ds \land \,$ $\ds y_1$ $=$ $\ds y_2$

$\blacksquare$