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Math Help - Negating normal form

  1. #1
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    Negating normal form

    Hi!

    I have a formula given like this(the first V is supposed to be up side down, /\ as in "and" not "or"):

     \sim (\sim PV  \sim(\sim\sim Q \vee \sim R)) \\

    How do I approach this to "simplifie" it (negating normal form).


    Any help would be greatly appriciated!

    Eg.
     \sim (P=>Q) \\

    Which equalies/on a simpler form using De Morgan's law

     P.\sim Q \\

    . is "and" as in upside down V
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  2. #2
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    Hello, jokke22!

    Simplify: .  \sim \bigg[\sim\!P\: \wedge   \sim(\sim\sim\! Q \:\vee \sim\!R)\bigg] \\

    We have: . \sim\bigg[\sim\!P\:\wedge \sim(\sim\sim\!Q\:\vee \sim\!R)\bigg]

    . . . . . . = \;\;\sim\bigg[P\:\wedge \sim(Q \:\vee \sim\!R)\bigg] . . double negative

    . . . . . . = \;\;\sim\bigg[P\:\wedge \sim\!Q\:\wedge R\bigg] . . . . DeMorgan's Law

    . . . . . . =\;\;\sim\!P \:\vee Q \:\vee \sim\!R . . . . . DeMorgan's Law

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  3. #3
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    This is what you posted.
    \begin{gathered}<br />
  \neg \left[ {P \vee \neg \left( {\neg \neg Q \vee \neg R} \right)} \right] \hfill \\<br />
  \neg \left[ {P \vee \neg \left( {Q \vee \neg R} \right)} \right] \hfill \\<br />
  \neg \left[ {P \vee \left( {\neg Q \wedge R} \right)} \right] \hfill \\<br />
  \left[ {\neg P \wedge \neg \left( {\neg Q \wedge R} \right)} \right] \hfill \\<br />
  \left[ {\neg P \wedge \left( {Q \vee \neg R} \right)} \right] \hfill \\ <br />
\end{gathered}

    Did you post it correctly?
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  4. #4
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    Appriciate it mate! Thank you!


    Since I have your attention, im struggling a bit with this one as well:

    \sim\bigg[(P->Q)->\sim(R->S)\bigg]
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  5. #5
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    Quote Originally Posted by Plato View Post
    This is what you posted.
    \begin{gathered}<br />
  \neg \left[ {P \vee \neg \left( {\neg \neg Q \vee \neg R} \right)} \right] \hfill \\<br />
  \neg \left[ {P \vee \neg \left( {Q \vee \neg R} \right)} \right] \hfill \\<br />
  \neg \left[ {P \vee \left( {\neg Q \wedge R} \right)} \right] \hfill \\<br />
  \left[ {\neg P \wedge \neg \left( {\neg Q \wedge R} \right)} \right] \hfill \\<br />
  \left[ {\neg P \wedge \left( {Q \vee \neg R} \right)} \right] \hfill \\ <br />
\end{gathered}

    Did you post it correctly?
    Yes that is correct, EXCEPT for the first \vee that should be \wedge, sorry for my bad typing here, not familiar with all the commands yet!
    Last edited by jokke22; February 26th 2009 at 02:26 AM.
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  6. #6
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    Quote Originally Posted by jokke22 View Post
    Appriciate it mate! Thank you!


    Since I have your attention, im struggling a bit with this one as well:

    \sim\bigg[(P->Q)->\sim(R->S)\bigg]

    ~[(p--->q)----->~(r---->s)] =

    =~[~(~pvq) v ~(~r v s)]=..................by material implication:P---->Q=~P V Q

    = (~pvq) ^ (~r v s)=.........................by de morgan...


    =(p---->q) ^ (r----->s).....................by material implication again
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  7. #7
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    Hello again, jokke22!

    \sim\bigg[(P \to Q)\;\to\; \sim\!(R \to S)\bigg]

    \sim\bigg[(\sim\! P \vee Q) \;\to\;\sim(\sim\! R \vee S)\bigg] . . def. of Implication

    \sim\bigg[(\sim\!P \vee Q) \;\to\;  (R\: \wedge \sim\!S)\bigg]. . . .DeMorgan

    \sim\bigg[\sim(\sim\!P \vee Q) \;\vee\; (R \:\wedge \sim\!S)\bigg] . . def. of Implication

    . . . (\sim\!P \vee Q) \;\wedge\; \sim(R \:\wedge \sim\!S) . . .DeMorgan

    . . . (\sim\!P \vee Q) \;\wedge\; (\sim\!R \vee S) . . . . DeMorgan

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  8. #8
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    Quote Originally Posted by Soroban View Post
    Hello again, jokke22!


    \sim\bigg[(\sim\! P \vee Q) \;\to\;\sim(\sim\! R \vee S)\bigg] . . def. of Implication

    \sim\bigg[(\sim\!P \vee Q) \;\to\;  (R\: \wedge \sim\!S)\bigg]. . . .DeMorgan

    \sim\bigg[\sim(\sim\!P \vee Q) \;\vee\; (R \:\wedge \sim\!S)\bigg] . . def. of Implication

    . . . (\sim\!P \vee Q) \;\wedge\; \sim(R \:\wedge \sim\!S) . . .DeMorgan

    . . . (\sim\!P \vee Q) \;\wedge\; (\sim\!R \vee S) . . . . DeMorgan

    Thank you yet again! As to benes!
    Now I see how these can be solved by adding these laws one by one to simplify them!
    As to the laws and such do you know of any great resources which I can look at as well?
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  9. #9
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    As for this:

    \sim\bigg[  \sim (\ P \to Q) \vee\sim R \bigg]

    \sim\bigg[  \sim (\sim \ P \vee Q) \vee\sim R \bigg]

    \sim\bigg[  P \vee (Q \vee\sim R) \bigg]

    \bigg[  \sim P \wedge (Q \vee\sim R) \bigg]

    \sim P \wedge (\sim Q \wedge R)


    Gave it a try but pretty sure this one is wrong though...
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