Solution:
(i)
$f: N \rightarrow N: f(x)=x^{3}$ is one – one into.
$f(x)=x^{3}$
As the function $f(x)$ is monotonically increasing from the domain $N \rightarrow N$ $\therefore f(x)$ is one -one
Range of $f(x)=(-\infty, \infty) \neq N$ (codomain)
$\therefore f(x)$ is into
$\therefore f: \mathbb{N} \rightarrow \mathrm{N}: f(x)=x^{2}$ is one – one into.
(ii)
$f: Z \rightarrow Z: f(x)=x^{3}$ is one – one into
$f(x)=x^{3}$
As the function $f(x)$ is monotonically increasing from the domain $Z \rightarrow Z$ $\therefore f(x)$ is one -one
Range of $f(x)=(-\infty, \infty) \neq Z$ (codomain)
$\therefore f(x)$ is into
$\therefore f: Z \rightarrow Z: f(x)=x^{3}$ is one $-$ one into.