Surjection/Examples/Negative Function on Integers

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Example of Surjection

Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:

$\forall x \in \Z: \map f x = -x$

Then $f$ is a surjection.


For $f$ to be a surjection, it is necessary that:

$\forall y \in \Z: \exists x \in \Z: -x = y$

This is the case by setting $ x = -y$, which is an integer since $y$ is an integer.

Thus $f$ is a surjection by definition.