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Proving De Morgan's Laws

  1. May 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove De Morgan's Laws (only (A U B)' = A' n B' part)

    2. Relevant equations

    (A U B)' = A' n B'

    3. The attempt at a solution

    I used this
    x does not belong to (A U B)
    x belongs to A' and B'
    x belongs to A' n B'

    now opposite

    x belongs to A' n B'
    x belogns to A' and B'
    x does not belong to A and B

    and that's it.
    how can this x does not belong to A and B can be changed into
    x does not belong to (A U B)' ?

    all these proving things are confusing..they are too obvious and that's why i can not do this properly how silly..
    Any tips that can be used when solving this kind of questions?

    Thankyou!
     
  2. jcsd
  3. May 8, 2009 #2

    dx

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    [tex] (x \in A' \cap B') \Rightarrow \neg(x \in A)\wedge \neg (x \in B) = \neg[ (x \in A) \vee (x \in B) ] \Rightarrow (x \in (A \cup B)') [/tex]

    Does that make sense?
     
  4. May 8, 2009 #3
    @dx - you are using demorgans law to prove demorgans law....

    You used that form of the law that applies to mathematical logic, to get the same law in a different form in set theory..

    So that proof is void
     
  5. May 8, 2009 #4
    @dx - you are using demorgans law to prove demorgans law....

    You used that form of the law that applies to mathematical logic, to get the same law in a different form in set theory..

    So that proof is void
     
  6. May 8, 2009 #5

    dx

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    How can you prove anything if you're not allowed to use logic?
     
  7. May 8, 2009 #6
    you can use logic. What i was talking about was a branch of mathematics - mathematical logic. You have to start from the axioms and prove the law that u have used in your proof. Start from the axioms of 'Mathematical Logic', and use those to prove it. It can be proved, but u must do it. Once that is done, rest follows

    What you have done is use DeMorgan's Law in ML to prove the same in set theory. That is where u are wrong.
     
  8. May 8, 2009 #7
    you can use logic. What i was talking about was a branch of mathematics - mathematical logic. You have to start from the axioms and prove the law that u have used in your proof. Start from the axioms of 'Mathematical Logic', and use those to prove it. It can be proved, but u must do it. Once that is done, rest follows

    What you have done is use DeMorgan's Law in ML to prove the same in set theory. That is where u are wrong.
     
  9. May 9, 2009 #8
    what is this \neg stand for?
     
  10. May 9, 2009 #9

    HallsofIvy

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    Very good but more precisely: "Let x belong to (A U B)'. Then x does not belong to A U B and so is not in A and is not in B. Therefore x is in A' and x is in B'. Then x is in A' n B'.

    Let x be in A' n B'. Then x is in A' and in B'. Since x is in A', x is not in A. Since x is in B', x is not in B. If x were in A U B, it would have to be in either A or B. Since it is not, it is not in A U B and so is in A' U B'.

     
    Last edited: May 9, 2009
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