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If f: A → B is a function and |A| = 5, |B| = 3, what is the maximum number of di

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Question: If f: A → B is a function and |A| = 5, |B| = 3, what is the maximum number of distinct functions f?

Options:

  1. 3^5
  2. 5^3
  3. 15
  4. 8

Correct Answer: 3^5

Solution: 

The number of distinct functions from set A to set B is given by |B|^|A|. Here, |B| = 3 and |A| = 5, so the maximum number of distinct functions is 3^5 = 243.

If f: A → B is a function and |A| = 5, |B| = 3, what is the maximum number of di

Practice Questions

Q1
If f: A → B is a function and |A| = 5, |B| = 3, what is the maximum number of distinct functions f?
  1. 3^5
  2. 5^3
  3. 15
  4. 8

Questions & Step-by-Step Solutions

If f: A → B is a function and |A| = 5, |B| = 3, what is the maximum number of distinct functions f?
  • Step 1: Understand that a function f: A → B means that each element in set A is assigned to an element in set B.
  • Step 2: Identify the sizes of the sets. Here, |A| = 5 means there are 5 elements in set A, and |B| = 3 means there are 3 elements in set B.
  • Step 3: Realize that for each element in set A, you can choose any of the 3 elements in set B to map to.
  • Step 4: Since there are 5 elements in A and each can map to any of the 3 elements in B, you have 3 choices for the first element, 3 choices for the second element, and so on.
  • Step 5: Calculate the total number of distinct functions by multiplying the number of choices for each element in A. This is done by raising the number of choices (3) to the power of the number of elements in A (5).
  • Step 6: Perform the calculation: 3^5 = 3 * 3 * 3 * 3 * 3 = 243.
  • Step 7: Conclude that the maximum number of distinct functions from set A to set B is 243.
  • Functions and Cardinality – Understanding how to calculate the number of distinct functions based on the sizes of the domain and codomain.
  • Exponential Growth – Recognizing that the number of functions grows exponentially with the size of the domain and codomain.
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