# Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as

Question:

Consider $f:\{1,2,3\} \rightarrow\{a, b, c\}$ and $g:\{a, b, c\} \rightarrow\{$ apple, ball, catt defined as $f(1)=a, f(2)=b, f(3)=c, g(\mathrm{a})=$ apple, $g(\mathrm{~b})=$ ball and $g(\mathrm{c})=$ cat. Show that $f, g$ and goo are

invertible. Find $f^{-1}, g^{-1}$ and gof $^{-1}$ and show that $(g \circ f)^{-1}=f^{-1} \circ g^{-1}$.

Solution:

$f=\{(1, a),(2, b),(3, c)\}$ and $g=\{(a$, apple $),(b$, ball $),(c$, cat $)\}$

Clearly, $f$ and $g$ are bijections.

So, $f$ and $g$ are invertible.

Now,

$f^{-1}=\{(a, 1),(b, 2),(c, 3)\}$ and $g^{-1}=\{($ apple,$a)$, (ball, $b$ ), (cat, $\left.c)\right\}$

So, $f^{-1} o g^{-1}=\{($ apple, 1$),($ ball, 2$),($ cat, 3$)\} \quad \ldots(1)$

$f:\{1,2,3\} \rightarrow\{a, b, c\}$ and $g:\{a, b, c\} \rightarrow\{$ apple, ball, cat $\}$

So, $g o f:\{1,2,3\} \rightarrow\{$ apple, ball, cat $\}$

$\Rightarrow(g \circ f)(1)=g(f(1))=g(a)=$ apple

$(g o f)(2)=g(f(2))=g(b)=$ ball

and $(g o f)(3)=g(f(3))=g(c)=$ cat

$\therefore g o f=\{(1$, apple $),(2$, ball $),(3$, cat $)\}$

Clearly, gofis a bijection.

So, gof is invertible.

$(\text { gof })^{-1}=\{($ apple, 1$),($ ball, 2$),($ cat, 3$)\}$             ...(2)

From $(1)$ and $(2)$, we get:

$(g o f)^{-1}=f^{-1} o g^{-1}$