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
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 \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33d7469d-24b0-4323-9278-1b0790a1937b%2Fa9cc6756-f2ad-4a6d-b0fc-173c5b3d67c3%2Fmxs8c2h_processed.png&w=3840&q=75)
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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