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Math Help - Proving Isomorphism

  1. #1
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    Proving Isomorphism

    Proving Isomorphism-mathproblem.png


    THe problem is attached. Anyone have any ideas?
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  2. #2
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    Re: Proving Isomorphism

    Quote Originally Posted by nexttime35 View Post
    Click image for larger version. 

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    THe problem is attached. Anyone have any ideas?
    The title is misleading. All that is required is to show a bijection between the sets.

    If \alpha\in X^{\{1,2\}} then \exists x_1\in X~\&~\exists x_2\in X such that \alpha=\{(1,x_1),(2,x_2)\}.

    Now (x_1,x_2)\in X\times X. You should be able to complete the question.
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    Re: Proving Isomorphism

    Now to finish this question, to show a bijection, I will need to show the injection (1-1) and onto-ness (surjection). In order for a function to be 1-1, for all x1,x2 in X x X, f(x1)=f(x2) implies that x1 = x2. And in order for the function to be onto, for all elements, say (x1,x2) in the codomain X x X , there exists (f(1),f(2)) in the domain such that f(f(1),f(2)) = (x1, x2).

    Would that be correct?
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    Re: Proving Isomorphism

    Quote Originally Posted by nexttime35 View Post
    Now to finish this question, to show a bijection, I will need to show the injection (1-1) and onto-ness (surjection). In order for a function to be 1-1, for all x1,x2 in X x X, f(x1)=f(x2) implies that x1 = x2. And in order for the function to be onto, for all elements, say (x1,x2) in the codomain X x X , there exists (f(1),f(2)) in the domain such that f(f(1),f(2)) = (x1, x2).
    Would that be correct?
    Suppose that \alpha\in X^{\{1,2\}} define \Phi(\alpha)=(\alpha(1),\alpha(2))~.

    Now is it clear that \Phi:X^{\{1,2\}}\to X\times X~?

    If \Phi(\alpha)=\Phi(\beta) can you show that \alpha=\beta~?

    If (a,b)\in X\times X can you show \exists\gamma\in X^{\{1,2\}} such that \Phi(\gamma)=(a,b)~?
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    Re: Proving Isomorphism

    So, Sigma(alpha) = Sigma (Beta) implies that (alpha(1),alpha(2)) = (beta(1), beta(2)), which implies that alpha = beta.

    onto: So, let (a,b) in X x X be arbitrary and let (gamma) = sigma(alpha). Then sigma(gamma)= (gamma(1), gamma(2)) = (a,b)?

    Does that follow? Thanks for walking this through, I am definitely learning the process here.
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    Re: Proving Isomorphism

    Quote Originally Posted by nexttime35 View Post
    onto: So, let (a,b) in X x X be arbitrary and let (gamma) = sigma(alpha). Then sigma(gamma)= (gamma(1), gamma(2)) = (a,b)
    Let \sigma=\{(1,a),(2,b)\} then is it clear to you that \sigma\in X^{\{1,2\}}~?

    Then definition \Phi(\sigma)=(a,b)~.
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