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Math Help - One to One functions and Inverses.

  1. #1
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    One to One functions and Inverses.

    If V = {0, 1, 2}. Describe all functions T: V --> V for which T(V) = V. There are six of them. Indicate which ones are one to one and give their inverses.

    So I believe something like T1(0) = 0, T2(1) = 1 and T3(2) = 2 will be the first three functions? How would you get the other three functions?
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  2. #2
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    No, you've described, essentially, only one of the functions, and you have 5 left. To define a function on V, you must describe where each element gets mapped to by T. You could describe your first function there as (0,1,2) -> (0,1,2). Another function would be (0,1,2) -> (0,2,1). Do you know what the name is of this mathematical idea? That will help you get all the desired functions.
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  3. #3
    Member HappyJoe's Avatar
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    Also, the condition T(V) = V means that the function T is surjective. It is a theorem, which you probably may find in your text book, that a surjective map from a finite set to itself is automatically one to one.
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  4. #4
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    And, indeed, that means that the 6 functions are the 6 permutations on{0, 1, 2}.
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  5. #5
    A Plied Mathematician
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    Reply to HallsofIvy:

    I was kinda hoping hashshashin715 would see that on his own...
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