State with reason whether following functions have inverse

Solution:

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

$f=\{(1,10),(2,10),(3,10),(4,10)\}$

From the given definition of f, we can see that f is a many one function as: f(1) = f(2) = f(3) = f(4) = 10

∴f is not one-one.

Hence, function f does not have an inverse.

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

$g=\{(5,4),(6,3),(7,4),(8,2)\}$

From the given definition of $g$, it is seen that $g$ is a many one function as: $g(5)=g(7)=4$.

∴g is not one-one,

Hence, function g does not have an inverse.

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

It is seen that all distinct elements of the set {2, 3, 4, 5} have distinct images under h.

∴Function h is one-one.

Also, h is onto since for every element y of the set {7, 9, 11, 13}, there exists an element x in the set {2, 3, 4, 5}such that h(x) = y.

Thus, h is a one-one and onto function. Hence, h has an inverse.