# Question ea4c3

Nov 6, 2015

By writing out a few terms for the function z, we see that
z={(1,1),(2,2),(3,8),(4,13),(5,8),(6,13),(7,8), 8,13),........}#

By definition, a function $f : X \to Y$ is onto (surjective) $\iff \forall y \in Y , \exists X \in X$ such that $f \left(x\right) = y$.

So in this case for example, $\exists 4 \in \mathbb{Z}$ such that $x \left(x\right) \ne 4 \forall x \in \mathbb{Z}$.
$\therefore z$ is not onto.

By definition, a function $f : X \to Y$ is 1 -1 (injective) $\iff f \left({x}_{1}\right) = f \left({x}_{2}\right) \implies {x}_{1} = {x}_{2}$.

So in this case for example, $3 \ne 5$ but $z \left(3\right) = z \left(5\right) = 8$.

$\therefore z$ is not 1 - 1.