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Math Help - preimage and image stuff

  1. #1
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    preimage and image stuff

    how do you prove the preimage of the image of the domain is equal to the domain and how do you prove the image of the preimage of the codomain is the codomain
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  2. #2
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    please help
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by leinadwerdna View Post
    how do you prove the preimage of the image of the domain is equal to the domain and how do you prove the image of the preimage of the codomain is the codomain
    Problem: Let \phi:X\mapsto Y. Prove that X=\phi^{-1}\left[\phi\left(X\right)\right]

    Proof: Let x\in\phi^{-1}\left[\phi\left(X\right)\right]. Then \phi(x)\in \phi\left(X\right)\implies x\in X. Therefore \phi^{-1}\left[\phi\left(X\right)\right]\subset X. Now let x\in X, \phi(x)\in\phi\left(X\right). Which means that x\in\phi^{-1}\left[\phi\left(X\right)\right]. Therefore X\subset\phi^{-1}\left[\phi\left(X\right)\right]. The conclusion follows.

    I hope that is rigorous enough.

    The second is similar.
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