Question

If $A=\{1,2,3\}$, show that a one-one function $f:A\to A$ must be onto.

Answer 1

We have, $A=\{1,2,3\}$

Function $f:A\to A$

$\therefore$ Function $f:\{1,2,3\}\to \{1,2,3\}$

Since $f$ is one-one.

$\left(i\right)$ $f\left(1\right)=1,f\left(2\right)=2,f\left(3\right)=3$
$\left(ii\right)$ $f\left(1\right)=1,f\left(2\right)=3,f\left(3\right)=2$
$\left(iii\right)$ $f\left(1\right)=2,f\left(2\right)=3,f\left(3\right)=1$
$\left(iv\right)$ $f\left(1\right)=2,f\left(2\right)=1,f\left(3\right)=3$
$\left(v\right)$ $f\left(1\right)=3,f\left(2\right)=1,f\left(3\right)=2$

$\left(vi\right)$ $f\left(1\right)=3,f\left(2\right)=2,f\left(3\right)=1$

Hence every element $1,2,3$ has either of image $1,2,3$ and that image is unique.

In each case, every image has a corresponding element.

Hence, function $f:A\to A$ is onto.