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Math Help - Showing That it is onto?

  1. #1
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    Showing That it is onto?

    Define a binary operation on the rationals so that the map defined by is an isomorphism between sets with binary operations. Prove that is an isomorphism.

    THIS IS WHAT I HAVE TO DO
    "We now give an outline to show two binary structures (S,*) and (S,*') are isomorphic.

    To prove it is an isomorphism i have to:
    1) Define the function \phi that gives the isomorphism of S with S'. Now this means that we have to describe, in some fashion, what \phi (s) is for every s \in S.

    2) Show that \phi is a one-to-one function. That is, suppose that \phi (x) = \phi (y) in S' and deduce from this that x = y in S.

    3) Show that \phi is onto S'. that is, suppose that s' \in S' is given and show that there does exist s \in S such that \phi(s) = s'

    4) Show that \phi(x*y) = \phi(x) *' \phi(y) for all x,y \in S. This is just a question of computation. Compute both sides of the equation and see whether they are the same."

    ----------------------------------------------------------------

    I have defined u \bullet v = u + v + \frac{1}{7}
    1) The function has already been defined \phi(x) = 7x + 1

    2) Lets Suppose \phi(x) = \phi(y)
    then 7x + 1 = 7y + 1
    7x = 7y
    x = y
    Therefore it is one-to-one

    3) I dont even know were to start

    4) \phi(u \bullet v) = \phi(u) + \phi(v)
    7(u \bullet v) + 1 = 7u + 1 + 7v + 1
    7(u \bullet v)= 7(u + v + \frac{1}{7})
    u \bullet v= u + v + \frac{1}{7}

    So what i need help is part 3) and i would like some1 to tell me if i had done the other parts correctly?
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  2. #2
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    3) Let s'\in S'=\mathbb{Q}. The function \phi is onto if we can find x so that \phi(x)=7x+1=s'. But that easy x should be the solution to 7x+1=s', i.e., x=\frac{s-1}{7} and it is in Q. Hence \phi is onto


    For the whole thing your idea is OK. But to be in order you need to
    1) define what u*v means (well you guess this from your calculation in 4)
    2) shows \phi(u*v)=\phi(u)+\phi(v) (i.e, \phi is a homomorphism). This calculation guides you to answer 1, but in writing you need to answer 1 first and somehow you you define u*v out of thin air.
    3) show that it is injective
    4) show it is onto.
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  3. #3
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    Thanks, i think i understand now.
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