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Math Help - Sets: Direct Proof

  1. #1
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    Sets: Direct Proof

    Show that is false for all sets A, B and C using direct proof.

    I'm not sure how to finish off my proof... can someone please help

    If we were to prove that the statement is true then we would need to show that if then and if then

    To formalise this we denote the following:

    Let P(x) denote the proposition function "" and Q(x) denote the proposition function ""

    Thus we need to show that and

    Let us first focus on

    Because we are using a direct proof let us assume the hypothesis P(x) is true. If we can show that Q(x) is false then the universally quantified statement is false and we have complete our proof.

    Since P(x) is assumed to be true then can be interpreted as or .

    is only true if:

    1. is true and is true

    2. is true and is false

    3. is false and is true

    Now Q(x) is interpreted as and

    Now what...?
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  2. #2
    A Plied Mathematician
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    (A\cap B)\cup C=A\cap(B\cup C)=A=B=C if A=B=C. You're going to need conditions on the three sets in order to make it false. It's certainly not going to hold for all sets!
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  3. #3
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    Oh yes, I see, thanks very much Ackbeet!!
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  4. #4
    A Plied Mathematician
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    You're very welcome. Have a good one!
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  5. #5
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    Do you mean "false for all sets" or "not true for all sets"?

    The first is, as Akbeet said, not a true statement so you cannot prove it.

    But the second, that there exist some sets for which (A\cap B)\cup C\ne A\cap(B\ cup C), is true. A counter example is the simplest way to show that- exhibit one of those sets.
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