Question

If A = {1, 2, 3, 4} and B = {a, b, c, d}, define any four bijections from A to B. Also give their inverse functions.

Answer 1

$$\begin{gathered} f_1=\left\{\left(1,a\right),\left(2,b\right),\left(3,c\right),\left(4,d\right)\right\}\Rightarrow {f_1}^{-1}=\left\{\left(a,1\right),\left(b,2\right),\left(c,3\right),\left(d,4\right)\right\} \\ {f}_2=\left\{\left(1,b\right),\left(2,a\right),\left(3,c\right),\left(4,d\right)\right\}\Rightarrow {f_2}^{-1}=\left\{\left(b,1\right),\left(a,2\right),\left(c,3\right),\left(d,4\right)\right\} \\ {f}_3=\left\{\left(1,a\right),\left(2,b\right),\left(4,c\right),\left(3,d\right)\right\}\Rightarrow {f_3}^{-1}=\left\{\left(a,1\right),\left(b,2\right),\left(c,4\right),\left(d,3\right)\right\} \\ {f}_4=\left\{\left(1,b\right),\left(2,a\right),\left(4,c\right),\left(3,d\right)\right\}\Rightarrow {f_4}^{-1}=\left\{\left(b,1\right),\left(a,2\right),\left(c,4\right),\left(d,3\right)\right\} \end{gathered}$$
Clearly, all these are bijections because they are one-one and onto.