Question

Given 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.

Answer 1

(i) Let $f:N\to N$, be a mapping defined by f(x)=2x
which is one-one
For $f\left(x_1\right)=f\left(x_2\right)$
$\Rightarrow 2x_1=2x_2{x}_1=x_2$
Further f is not onto. as for $1\in N$, there does not exist any x in 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. $f$ is not one-one as f(1)=f(2)=1. But f is onto.

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