State with reasons whether the following functions have inverse:

Solution:

(i) $f:\{1,2,3,4\} \rightarrow\{10\}$ with $f=\{(1,10),(2,10),(3,10),(4,10)\}$

We have:

$f(1)=f(2)=f(3)=f(4)=10$

$\Rightarrow f$ is not one-one.

$\Rightarrow f$ is not a bijection.

So, f does not have an inverse.

(ii) $g:\{5,6,7,8\} \rightarrow\{1,2,3,4\}$ with $g=\{(5,4),(6,3),(7,4),(8,2)\}$

$g(5)=g(7)=4$

$\Rightarrow f$ is not one-one.

$\Rightarrow f$ is not a bijection.

So, $f$ does not have an inverse.

(iii) $h:\{2,3,4,5\} \rightarrow\{7,9,11,13\}$ with $h=\{(2,7),(3,9),(4,11),(5,13)\}$

Here, different elements of the domain have different images in the co-domain.

$\Rightarrow h$ is one-one.

Also, each element in the co-domain has a pre-image in the domain.

$\Rightarrow h$ is onto.

$\Rightarrow h$ is a bijection.

$\Rightarrow h$ has an inverse and it is given by

$h^{-1}=\{(7,2),(9,3),(11,4),(13,5)\}$