Question

Let $S=\{1,2,3\}$. Determine whether the functions $f:S\to S$ defined as below has inverse. Find $f^{-1},$ if it exists.
(i) $f=\left\{\right(1,1),(2,2),(3,3\left)\right\}$
(ii) $f=\left\{\right(1,2),(2,1),(3,1\left)\right\}$
(iii) $f=\left\{\right(1,3),(3,2),(2,1\left)\right\}$

Answer 1

Checking for one-one.
$f=\left\{\right(1,1),(2,2),(3,3\left)\right\}$
Since each element has distinct image, $f$ is one-one.

Checking for onto.
$f=\left\{\right(1,1),(2,2),(3,3\left)\right\}$
Since, for every image, there is a corresponding element,
$\therefore f$ is onto.

Finding inverse of $f$.
$\because f=\left\{\right(1,1),(2,2),(3,3\left)\right\}$ is both one-one and onto. So $f$ is invertible
$\therefore f^{-1}=\left\{\right(1,1),(2,2),(3,3\left)\right\}$

$f=\left\{\right(1,2),(2,1),(3,1\left)\right\}$
Since $2\&3$ have the same image $1$.
$f$ is not one-one.
Since, $f$ is not -ne-one, it's inverse does not exist.

Checking for one-one.
$f=\left\{\right(1,3),(3,2),(2,1\left)\right\}$
Each element has distinct image
$\therefore f$ is one-one.

Checking for onto.
$f=\left\{\right(1,3),(3,2),(2,1\left)\right\}$
For every image, there is a corresponding element.
$\therefore f$ is onto.

Finding inverse of $f$
Hence, $f^{-1}=\left\{\right(1,3),(3,2),(2,1\left)\right\}$