Functions
The number of onto functions from a set A containing m elements to a set B containing n elements (m ≥ n) is
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Solution
n! S(m,n)
- Combinatorial Logic: An onto function requires every element in B to have at least one pre-image.
- Stirling Numbers: S(m,n) represents the number of ways to partition a set of m elements into n non-empty, unlabelled subsets.
- Permutation: Since the n elements in the codomain are distinct (labelled), we multiply the partitions by n! to assign them to specific codomain elements.
- Formula: Total onto functions = n!×S(m,n).
- This can also be solved using the Principle of Inclusion-Exclusion.
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- functions
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
n! S(m,n)
- Combinatorial Logic: An onto function requires every element in B to have at least one pre-image.
- Stirling Numbers: S(m,n) represents the number of ways to partition a set of m elements into n non-empty, unlabelled subsets.
- Permutation: Since the n elements in the codomain are distinct (labelled), we multiply the partitions by n! to assign them to specific codomain elements.
- Formula: Total onto functions = n!×S(m,n).
- This can also be solved using the Principle of Inclusion-Exclusion.
This hard difficulty mathematics question is from the chapter sets, relations and functions, covering the topic of functions. It appeared in the 2025 exam.
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