Surjection/Examples/Half Even Zero Odd
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Example of Surjection
Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:
- $\forall x \in \Z: \map f x = \begin{cases} \dfrac x 2 & : x \text { even} \\ 0 & : x \text { odd} \end{cases}$
Then $f$ is a surjection.
Proof
Let $y \in \Z$ be an integer.
Consider the integer $x = 2 y$.
Then:
- $\exists x \in \Z: y = \dfrac x 2$
That is:
- $x = 2 y$
Thus $f$ is a surjection by definition.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $3$. Mappings: Exercise $2 \ \text {(iii) (b)}$