Find Number of One-One Functions from \(A\) to \(B\)

📝 Question

Let:

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

Find the total number of one-one (injective) functions from \(A\) to \(B\).

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✅ Solution

🔹 Step 1: Compare Sizes

\[ |A|=4,\quad |B|=3 \]

For a function to be one-one, each element of \(A\) must map to a distinct element of \(B\).

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🔹 Step 2: Key Concept

If \(|A| > |B|\), then it is impossible to assign distinct images to all elements of \(A\).

At least two elements of \(A\) will map to the same element of \(B\).

Thus, injective function cannot exist.

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🔹 Step 3: Conclusion

\[ \text{Number of one-one functions} = 0 \] —

🎯 Final Answer

\[ \boxed{0} \]

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🚀 Exam Shortcut

• If \(|A| > |B|\) ⇒ no one-one function
• Injection requires distinct outputs
• More inputs than outputs ⇒ impossible