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Math Help - Proving a function is one-to-one

  1. #1
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    Proving a function is one-to-one

    Problem:

    Prove that the function f: Q ---> R, f(a/b) = (2^a)(3^b) is one to one, assuming that gcd(a,b) = 1, that is the fraction a/b is reduced.



    I know f is one to one if it never maps 2 different elements to the same place, i.e. f(a) does not equal f(b) whenever a does not equal b.

    I have tried looking at how other problems are proved to be one to one and am attempting to do it in a similar way:

    Suppose f(a/b) = f(c/d)
    then (2^a)(3^b) = (2^c)(3^d)

    I'm not sure what to do though. I don't think I'm doing this right. Please help me with this problem.
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  2. #2
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    Quote Originally Posted by hannahs View Post
    Problem:

    Prove that the function f: Q ---> R, f(a/b) = (2^a)(3^b) is one to one, assuming that gcd(a,b) = 1, that is the fraction a/b is reduced.

    .
    Say f(a_1/b_1)=f(a_2/b_2)\implies 2^{a_1}3^{b_1}=2^{a_2}3^{b_2} by unique factorization of the integers a_1=a_2,b_1=b_2.
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