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Consider sets A and B with |A| = 12 and |B|| 8. How many functions f : A → B are surjective?

Consider sets A and B with |A| = 12 and |B|| 8. How many functions f : A → B are surjective?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Consider sets \( A \) and \( B \) with \(|A| = 12\) and \(|B| = 8\). How many functions \( f : A \to B \) are surjective? --- ### Explanation: To find the number of surjective functions from set \( A \) to set \( B \), we need to consider that a surjective function (or onto function) maps every element of \( B \) to at least one element in \( A \). Given: - \(|A| = 12\): Set \( A \) has 12 elements. - \(|B| = 8\): Set \( B \) has 8 elements. If \( f : A \to B \) is surjective, every element in \( B \) must be an image of some element in \( A \). ### Calculating Surjective Functions: 1. **Total Functions \( A \to B \):** Each of the 12 elements in \( A \) can be mapped to any of the 8 elements in \( B \). Thus, the total number of functions from \( A \) to \( B \) is: \[ 8^{12} \] 2. **Applying Inclusion-Exclusion Principle:** To find the number of surjective functions, we can use the principle of inclusion-exclusion. The formula for the number of surjective functions from a set with \( m \) elements to a set with \( n \) elements is: \[ n! \left( \sum_{i=0}^{n} \frac{(-1)^i}{i!} \left( n-i \right)^m \right) \] For this specific problem: - \( m = 12 \) - \( n = 8 \) Substituting these values into the formula gives us the count of surjective functions. ### Conclusion: By using the principle of inclusion-exclusion, one can calculate the exact number of surjective functions, ensuring that each element in the codomain \( B \) has a preimage in the domain \( A \).
Transcribed Image Text:**Problem Statement:** Consider sets \( A \) and \( B \) with \(|A| = 12\) and \(|B| = 8\). How many functions \( f : A \to B \) are surjective? --- ### Explanation: To find the number of surjective functions from set \( A \) to set \( B \), we need to consider that a surjective function (or onto function) maps every element of \( B \) to at least one element in \( A \). Given: - \(|A| = 12\): Set \( A \) has 12 elements. - \(|B| = 8\): Set \( B \) has 8 elements. If \( f : A \to B \) is surjective, every element in \( B \) must be an image of some element in \( A \). ### Calculating Surjective Functions: 1. **Total Functions \( A \to B \):** Each of the 12 elements in \( A \) can be mapped to any of the 8 elements in \( B \). Thus, the total number of functions from \( A \) to \( B \) is: \[ 8^{12} \] 2. **Applying Inclusion-Exclusion Principle:** To find the number of surjective functions, we can use the principle of inclusion-exclusion. The formula for the number of surjective functions from a set with \( m \) elements to a set with \( n \) elements is: \[ n! \left( \sum_{i=0}^{n} \frac{(-1)^i}{i!} \left( n-i \right)^m \right) \] For this specific problem: - \( m = 12 \) - \( n = 8 \) Substituting these values into the formula gives us the count of surjective functions. ### Conclusion: By using the principle of inclusion-exclusion, one can calculate the exact number of surjective functions, ensuring that each element in the codomain \( B \) has a preimage in the domain \( A \).
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