Bijection/Examples/Negative Functions

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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
\(\, \ds \land \, \) \(\ds y_1\) \(=\) \(\ds y_2\)


$\blacksquare$


Sources