# If A = {1, 2, 3, 4} and B = {a, b, c, d},

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.

Solution:

$f_{1}=\{(1, a),(2, b),(3, c),(4, d)\} \Rightarrow f_{1}^{-1}=\{(a, 1),(b, 2),(c, 3),(d, 4)\}$

$f_{2}=\{(1, b),(2, a),(3, c),(4, d)\} \Rightarrow f_{2}^{-1}=\{(b, 1),(a, 2),(c, 3),(d, 4)\}$

$f_{3}=\{(1, a),(2, b),(4, c),(3, d)\} \Rightarrow f_{3}^{-1}=\{(a, 1),(b, 2),(c, 4),(d, 3)\}$

$f_{4}=\{(1, b),(2, a),(4, c),(3, d)\} \Rightarrow f_{4}^{-1}=\{(b, 1),(a, 2),(c, 4),(d, 3)\}$

Clearly, all these are bijections because they are one-one and onto.