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Math Help - Proof with sets

  1. #1
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    Proof with sets

    I have to finish this proof.

    Let S and T be sets.

    Let S\oplus T be the symmetric difference of those sets.

    THM: \forall ~\mbox{sets} ~ A,B,C, ~\mbox{if}~ A \oplus C = B\oplus C,~\mbox{then}~A=B.


    Prove the above theorem by filling in the missing _____.

    Proof(Direct)

    Assume 1.) _______
    Show 2.) _______

    (\subseteq) Let x \in A. Show x\in B.

    Case 1:

    3.) ___________________________________

    x \in B

    Case 2: x \in C

    4.) ___________________________________

    x \in B

    (\subseteq) (<-- should be backwards, but not sure how to do it in latex).

    Similarly

    Q.E.D




    So 1 and 2 are easy, it's 3 and 4 that are rough.

    1.) A \oplus C = B\oplus C

    2.) A=B
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  2. #2
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    Quote Originally Posted by alikation View Post
    I have to finish this proof.

    Let S and T be sets.

    Let S\oplus T be the symmetric difference of those sets.

    THM: \forall ~\mbox{sets} ~ A,B,C, ~\mbox{if}~ A \oplus C = B\oplus C,~\mbox{then}~A=B.


    Prove the above theorem by filling in the missing _____.

    Proof(Direct)

    Assume 1.) _______
    Show 2.) _______

    (\subseteq) Let x \in A. Show x\in B.

    Case 1: {\color{red}x \not\in C}

    3.) ___________________________________

    {\color{red}x \not\in C \Rightarrow x \in A - C \subset A \oplus C = B \oplus C \Rightarrow x \in B \oplus C }

    But since {\color{red}x \not\in C \Rightarrow x \not\in C-B \Rightarrow x \in B-C}

    x \in B

    Case 2: x \in C

    4.) ___________________________________

    {\color{red}x \not\in A \oplus C \Rightarrow x \not\in B \oplus C \Rightarrow x \in B \cap C \Rightarrow x \in B}

    x \in B

    (\subseteq) (<-- should be backwards, but not sure how to do it in latex).

    Similarly

    Q.E.D


    So 1 and 2 are easy, it's 3 and 4 that are rough.

    1.) A \oplus C = B\oplus C

    2.) A=B
    ..
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