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Math Help - Basic Set Theory

  1. #1
    Super Member Aryth's Avatar
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    Basic Set Theory

    Prove the following for all sets A, B, and C:

    A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)

    and

    A\cap B \subset A \subset A \cup C
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Basic Set Theory

    Quote Originally Posted by Aryth View Post
    Prove the following for all sets A, B, and C:

    A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)

    and

    A\cap B \subset A \subset A \cup C
    The first is known as Demorgan's law. Try thinking about it logically, it says that x\in A\text{ and }(x\notin B\text{ or }x\notin C)\leftrightarrow (x\in A\text{ and }x\notin B)\text{ and }(x\in A\text{ and }x\notin C). Can you take it from there?


    The second part is fairly standard. Where are you having trouble?
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  3. #3
    Super Member Aryth's Avatar
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    Re: Basic Set Theory

    Actually, that's as far as I got on the first one... I did the \rightarrow but I got stumped trying to do the \leftarrow after stating what you did.

    On the second one, I guess I just can't see it...

    I've never been that good at Set Theory, it slightly confuses me for some reason. I have some time off so I'm trying to study it, and it's not going so well. :/
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Re: Basic Set Theory

    Quote Originally Posted by Aryth View Post
    Actually, that's as far as I got on the first one... I did the \rightarrow but I got stumped trying to do the \leftarrow after stating what you did.

    On the second one, I guess I just can't see it...

    I've never been that good at Set Theory, it slightly confuses me for some reason. I have some time off so I'm trying to study it, and it's not going so well. :/
    Ok. So, let's start with the second one. In sentence form A\cap B\subseteq A says "If x is in A and in B then it is in A", obvious now, right? Now try converting the second half.
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    Super Member Aryth's Avatar
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    Re: Basic Set Theory

    Quote Originally Posted by Drexel28 View Post
    Ok. So, let's start with the second one. In sentence form A\cap B\subseteq A says "If x is in A and in B then it is in A", obvious now, right? Now try converting the second half.
    Oh, ok... I see it now. For some reason I wasn't thinking of it like that. Of course if x is in A then it is in A or C.
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