Question:
Let $S=\{a, b, c\}$ and $T=\{1,2,3\} .$ Find $\mathrm{F}^{-1}$ of the following functions $\mathrm{F}$ from $S$ to $T$, if it exists.
(i) $F=\{(a, 3),(b, 2),(c, 1)\}$
(ii) $F=\{(a, 2),(b, 1),(c, 1)\}$
Solution:
S = {a, b, c}, T = {1, 2, 3}
(i) F: S → T is defined as:
F = {(a, 3), (b, 2), (c, 1)}
$\Rightarrow F(a)=3, F(b)=2, F(c)=1$
Therefore, $\mathrm{F}^{-1}: T \rightarrow S$ is given by
$\mathrm{F}^{-1}=\{(3, a),(2, b),(1, c)\}$
(ii) $F: S \rightarrow T$ is defined as:
$F=\{(a, 2),(b, 1),(c, 1)\}$
Since F (b) = F (c) = 1, F is not one-one.
Hence, $\mathrm{F}$ is not invertible i.e., $\mathrm{F}^{-1}$ does not exist.