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Math Help - how are these two statments logically equivalent?

  1. #1
    Member Jskid's Avatar
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    how are these two statments logically equivalent?



    It's not apparent how these are logically equivalent, could someone please elaborate?
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  2. #2
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    Re: how are these two statments logically equivalent?

    the second statement should be written as

    ((p\wedge \neg q)\vee q)\wedge((p\wedge \neg q)\vee \neg p)

    this follows from the first statement using distributive law

    Boolean algebra (logic) - Wikipedia, the free encyclopedia
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  3. #3
    Member Jskid's Avatar
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    Re: how are these two statments logically equivalent?

    But wouldn't that distribute as [(p \vee q ) \wedge ( \neg q \vee q)] \wedge [(p \vee \neg p) \wedge ( \neg q \vee \neg p )] which isn't the same as the prior step
    Last edited by Jskid; October 3rd 2011 at 09:23 PM. Reason: forgot \neg
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  4. #4
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    Re: how are these two statments logically equivalent?

    yes you are right. you have further used the distributive law.
    further you can see that
    q\vee \neg q \;\;\mbox{and}\;\; p\vee \neg p

    are tautologies. so we can further simplify as

    (p\vee q)\wedge (\neg p \vee \neg q)
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  5. #5
    Member Jskid's Avatar
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    Re: how are these two statments logically equivalent?

    I couldn't see that b/c I was trying to think of how to apply the distributive law differently. I never occured to me that I needed to do another step (i.e. remove tautologies)
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  6. #6
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    Re: how are these two statments logically equivalent?

    do you understand now ?
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