Define Bijections from \(A\) to \(B\) and Find Their Inverses

📝 Question

Let:

\[ A=\{1,2,3,4\}, \quad B=\{a,b,c,d\} \]

Define any four bijections from \(A\) to \(B\). Also find their inverse functions.

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✅ Solution

A function is bijective if it is both one-one and onto. :contentReference[oaicite:0]{index=0}

🔹 Bijection 1

\[ f_1=\{(1,a),(2,b),(3,c),(4,d)\} \]

Inverse:

\[ f_1^{-1}=\{(a,1),(b,2),(c,3),(d,4)\} \]

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🔹 Bijection 2

\[ f_2=\{(1,b),(2,a),(3,c),(4,d)\} \]

Inverse:

\[ f_2^{-1}=\{(b,1),(a,2),(c,3),(d,4)\} \]

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🔹 Bijection 3

\[ f_3=\{(1,a),(2,b),(3,d),(4,c)\} \]

Inverse:

\[ f_3^{-1}=\{(a,1),(b,2),(d,3),(c,4)\} \]

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🔹 Bijection 4

\[ f_4=\{(1,b),(2,a),(3,d),(4,c)\} \]

Inverse:

\[ f_4^{-1}=\{(b,1),(a,2),(d,3),(c,4)\} \]

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🎯 Final Answer

Four bijections and their inverses are:

\[ \boxed{ \begin{aligned} f_1^{-1}&=\{(a,1),(b,2),(c,3),(d,4)\} \\ f_2^{-1}&=\{(b,1),(a,2),(c,3),(d,4)\} \\ f_3^{-1}&=\{(a,1),(b,2),(d,3),(c,4)\} \\ f_4^{-1}&=\{(b,1),(a,2),(d,3),(c,4)\} \end{aligned} } \]

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🚀 Exam Shortcut

• Just rearrange elements of set \(B\)
• Each element must appear exactly once ⇒ bijection
• Inverse = reverse each ordered pair
• Total bijections = \(4! = 24\)