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.

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 $\map h {z_1} = \map h {z_2}$.