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Math Help - Prove propositions are logically equivalent

  1. #1
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    Prove propositions are logically equivalent

    I've got these questions and I've given it some thought but I'm not how to move forward. The question is.

    Let A and B be two sets. Prove the following propositions are logically equivalent.
    a) B is a subset of A
    b) A intersection of B = B
    c) A union of B = A

    I was thinking to convert this notation into proposition notation (and, or, not, etc..) and then use thruth tables to prove they are the same. The reason I cant move forward with this question this way is that I have no idea how set theory notation can be changed to proposition notation, or even it this is the right way to answer a question life this. Any help or resource you can point me to would be really great. thanks in advance.
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  2. #2
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    I will do one of these for you: c \Rightarrow a.
    Given that A\cup B=A. Suppose that x\in B.
    From the given we have x\in A\cup B=A so B\subseteq A.
    Now you do a \Rightarrow b~\&~b \Rightarrow c.
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  3. #3
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    clarification

    Thanks for your willingness to help...

    I'm starting to get it.. we are proving that question (a) is equivalent to question (b) and (b) is equivalent to question (c) , thus (c) is equivalent to question (a)

    I was thinking that questions (a), (b) and (c) where all serparate questions and unrelated and I needed to prove each statement serparately. I have attached these 2 approaches. Which method do you think is the correct way to proceed?

    Prove propositions are logically equivalent-unrelated.pdf
    Prove propositions are logically equivalent-related.pdf
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  4. #4
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    I have looked at both of the attachments.
    Frankly, I do not follow either.
    But the problem is clearly to show that those three are equivalent.
    a\to b\to c\to a.
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  5. #5
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    well looks like I have a lot more work to do...
    thank you.. :-)
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