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What is the opposite of this expression?

$p \land ( q \lor r )$

Please suggest any theorem as a starting point.

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If by "opposite" you mean "negation", look up De Morgan's Laws. –  Arturo Magidin Jan 4 '12 at 5:50
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Placing \$ and \$ around math expressions causes them to be rendered properly. –  Alex Becker Jan 4 '12 at 5:52
    
Yes. I do mean "negation". –  Dave Jan 4 '12 at 5:52
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1 Answer

up vote 2 down vote accepted

Keep in mind that $\lnot (A\land B) = (\lnot A)\lor (\lnot B)$ and $\lnot (A\lor B) = (\lnot A)\land (\lnot B)$. Using this, you can figure out $\lnot (p \land ( q \lor r ))$ as follows: $$\begin{eqnarray*}\lnot(p \land ( q \lor r )) &=& (\lnot p)\lor (\lnot(q\lor r))\\ &=& (\lnot p)\lor ((\lnot q)\land (\lnot r))\\\end{eqnarray*}$$ which you can distribute out if you wish.

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