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Math Help - Prove Boolean Expressions are equivelant

  1. #1
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    Unhappy Prove Boolean Expressions are equivelant

    Using the rules of boolean algebra prove (state which rule is used at each step):-

    <br />
\forall x in X[ p(x) \rightarrow \neg ( q(x) \wedge r(x) ) ] \equiv \neg \exists x in X[ p(x) \wedge q(x) \wedge r(x) ]<br />

    So far I have,

    <br />
\forall x in X[ \neg p(x) \lor \neg (q(x) \wedge r(x)) ]  Rewriting \rightarrow.<br />

    <br />
\neg \exists x in X[ p(x) \lor \neg (q(x) \wedge r(x) ) ]  Extended De Morgan (ii)<br />

    This is where I hit the brick wall. I've also tried doing the Extended De Morgan rule first but that doesn't seem right to me.

    any help appreciated.
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  2. #2
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    p \to \left( {q \wedge r} \right) \equiv \neg p \vee \left( {q \wedge r} \right) \equiv \neg \left( {p \wedge \neg (q \wedge r)} \right)
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  3. #3
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    Unhappy

    This has confused me even more??
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  4. #4
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    Unhappy

    I'm still struggling with this, not sure what rules you have used to get there.

    Help appreciated.
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  5. #5
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    I still need help with this if anyone can??
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  6. #6
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    Thumbs up

    I guess this is a trickier subject than I thought judging by the replies. Is there anyone out there who can check my final answer to this. I'm submitting it tomorrow.

    Step1
    ------
    Rewriting  \rightarrow

    <br />
\forall x \in X[ \neg p(x) \lor \neg ( q(x) \wedge r(x) ) ]<br />

    Step2
    ------
    De Morgan (ii)

    <br />
\forall x \in X[ \neg ( p(x) \wedge q(x) \wedge r(x) ) ]<br />

    Step3
    ------
    Extended De Morgan (ii)

    <br />
\neg \exists x \in X[ p(x) \wedge q(x) \wedge r(x) ]<br />

    This is the best I can come up with so it will have to do. Looks correct to me anyway
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