Give an example of a map
(i) which is one-one but not onto.
(ii) which is not one-one but onto.
(iii) which is neither one-one nor onto.

(i) Let $f:N \to N$, be a mapping defined by $f(x) = 2x$

which is one-one.
For $f\left( {{x_1}} \right) = f\left( {{x_2}} \right)$

$\Rightarrow$ $2{x_1} = 2{x_2}$

${x_1} = {x_2}$

Further $f$ is not onto, as for $1 \in N$, there does not exist any $x$ in $N$ such that $f(x) = 2x + 1$.

(ii) Let $f:N \to N$, given by $f(1) = f(2) = 1$ and $f(x) = x - 1$ for every $x > 2$ is onto but

not one-one. $f$ is not one-one as $f(1) = f(2) = 1$. But $f$ is onto.

(iii) The mapping $f:R \to R$ defined as $f(x) = {x^2}$, is neither one-one nor onto.