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Math Help - function composition and inverse function

  1. #1
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    Exclamation function composition and inverse function

    Let A={1,2,3,4,5} and B={6,7,8,9,10,11,12}.
    How many functions f:A->B are such that f -1({6,7,8})={1,2}?
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  2. #2
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    Quote Originally Posted by sbankica View Post
    Let A={1,2,3,4,5} and B={6,7,8,9,10,11,12}.
    How many functions f:A->B are such that f -1({6,7,8})={1,2}?
    Any function \phi :\{ 1,2\}  \mapsto \{ 6,7,8\} is such that \phi ^{ - 1} \{ 6,7,8\}  = \{ 1,2\} .
    There are 3^2 such functions. WHY?

    How many functions \{ 3,4,5\}  \mapsto \{ 9,10,11,12\}?

    Can you answer the question posted?
    Last edited by Plato; November 9th 2009 at 01:43 PM.
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  3. #3
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    4^3?
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  4. #4
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    so answer is: (3^2)(4^3)
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  5. #5
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    Quote Originally Posted by sbankica View Post
    so answer is: (3^2)(4^3)
    You got it!
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  6. #6
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    Quote Originally Posted by Plato View Post
    Any function \phi :\{ 1,2\} \mapsto \{ 6,7,8\} is such that \phi ^{ - 1} \{ 6,7,8\} = \{ 1,2\} .
    There are 3^2 such functions. WHY?

    How many functions \{ 3,4,5\} \mapsto \{ 9,10,11,12\}?

    Can you answer the question posted?
    PLATO, you have been so helpful.

    I see the following six functions:
    f(1)=6; f(1)=7; f(1)= 8; f(2)=6; f(2)=7; f(2)=8

    What are the other three functions?

    Thank you.
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  7. #7
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    Quote Originally Posted by oldguynewstudent View Post
    I see the following six functions:
    f(1)=6; f(1)=7; f(1)= 8; f(2)=6; f(2)=7; f(2)=8
    What are the other three functions?
    \left\{ {\left( {1,6} \right),\left( {2,6} \right)} \right\},\,\left\{ {\left( {1,7} \right),\left( {2,7} \right)} \right\},\,\left\{ {\left( {1,8} \right),\left( {2,8} \right)} \right\}
    \left\{ {\left( {1,6} \right),\left( {2,7} \right)} \right\},\,\left\{ {\left( {1,6} \right),\left( {2,8} \right)} \right\},\,\left\{ {\left( {1,7} \right),\left( {2,6} \right)} \right\}
    \left\{ {\left( {1,7} \right),\left( {2,8} \right)} \right\},\,\left\{ {\left( {1,8} \right),\left( {2,6} \right)} \right\}\;\& \,\left\{ {\left( {1,8} \right),\left( {2,7} \right)} \right\}
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  8. #8
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by Plato View Post
    \left\{ {\left( {1,6} \right),\left( {2,6} \right)} \right\},\,\left\{ {\left( {1,7} \right),\left( {2,7} \right)} \right\},\,\left\{ {\left( {1,8} \right),\left( {2,8} \right)} \right\}
    \left\{ {\left( {1,6} \right),\left( {2,7} \right)} \right\},\,\left\{ {\left( {1,6} \right),\left( {2,8} \right)} \right\},\,\left\{ {\left( {1,7} \right),\left( {2,6} \right)} \right\}
    \left\{ {\left( {1,7} \right),\left( {2,8} \right)} \right\},\,\left\{ {\left( {1,8} \right),\left( {2,6} \right)} \right\}\;\& \,\left\{ {\left( {1,8} \right),\left( {2,7} \right)} \right\}
    Thank you so much. I keep forgetting that the definition of a function maps every element in the domain to a unique element in the codomain.

    This will really help my understanding in the future.
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