Question

Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f⁻¹, g⁻¹ and gof⁻¹ and show that (gof)⁻¹ = f ⁻¹o g⁻¹.

Answer 1

$$\begin{gathered} f=\left\{\left(1,a\right),\left(2,b\right),\left(3,c\right)\right\}\text{and}g=\left\{\left(a,\text{apple}\right),\left(b,\text{ball}\right),\left(c,\text{cat}\right)\right\} \\ \text{Clearly,}f\text{and}g\text{are bijections.} \\ \text{So,}f\text{and}g\text{are invertible.} \\ \text{Now,} \\ {f}^{-1}=\left\{\left(a,1\right),\left(b,2\right),\left(c,3\right)\right\}\mathrm{and}{g}^{-1}=\left\{\left(\text{apple,}a\right),\left(\text{ball,}b\right),\left(\text{cat,}c\right)\right\} \\ \mathrm{So},f^{-1}o{g}^{-1}=\left\{\left(\text{apple, 1}\right),\left(\text{ball, 2}\right),\left(\text{cat, 3}\right)\right\}...\left(1\right) \\ f:\left\{1,2,3\right\}\to \left\{a,b,c\right\}\text{and}g:\left\{a,b,c\right\}\to \left\{\text{apple, ball, cat}\right\} \\ \text{So},gof:\left\{1,2,3\right\}\to \left\{\text{apple, ball, cat}\right\} \\ \Rightarrow \left(gof\right)\left(1\right)=g\left(f\left(1\right)\right)=g\left(a\right)=\text{apple} \\ \left(gof\right)\left(2\right)=g\left(f\left(2\right)\right)=g\left(b\right)=\text{ball,} \\ \mathrm{and}\left(gof\right)\left(3\right)=g\left(f\left(3\right)\right)=g\left(c\right)=\text{cat} \\ \therefore gof=\left\{\left(1,\text{apple}\right),\left(2,\text{ball}\right),\left(3,\text{cat}\right)\right\} \\ \text{Clearly,}gof\text{is a}biject\text{ion.} \\ \text{So,}gof\text{is invertible.} \\ {\left(gof\right)}^{-1}=\left\{\left(\text{apple, 1}\right),\left(\text{ball, 2}\right),\left(\text{cat, 3}\right)\right\}...\left(2\right) \\ \text{From}\left(1\right)\text{and}\left(2\right)\text{, we get:} \\ {\left(gof\right)}^{-1}=f^{-1}o{g}^{-1} \end{gathered}$$