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How many one-to-one functions f are possible?

  1. Mar 20, 2012 #1
    1. The problem statement, all variables and given/known data

    Suppose that |A| = n and |B| = m with n ≤ m. How many one-to-one functions f are possible with f: A → B?

    2. Relevant equations
    If |A| = |B| = m how many different bijections f: A → B are possible?
    Answer: m!

    3. The attempt at a solution
    I really do not know how to start off the question. If someone can help me get started into this equation that would be great!

    The relevant equation I did manage to get. but I do not know how to solve the problem when the size of A is not equal to the size of B.
     
  2. jcsd
  3. Mar 20, 2012 #2

    Dick

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    Science Advisor
    Homework Helper

    Well, you know f is a bijection into the subset of B given by f(A). Now you just have to figure out the number of ways to choose a subset of B of size |A|.
     
  4. Mar 20, 2012 #3
    Do you know how to compute the total number of functions from A to B? If you can do that, you can use the same type of reasoning for this problem, remembering that as you construct a function it must be 1-1.
     
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