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Problem 11: (De Morgan's Laws With Truth Tables)

Let $P$ and $Q$ be statements or open sentences. Start by completing the truth table below to give the truth values for $P\vee Q$, $\sim (P\vee Q)$, $(\sim P)\vee (\sim Q)$, and $(\sim P)\wedge (\sim Q).$ $$ \begin{array}{c|c|c|c|c|c|c|c} P&Q&P\vee Q&\sim(P\vee Q) &\sim P&\sim Q &(\sim P)\vee (\sim Q) & (\sim P)\wedge (\sim Q)\\\hline T&T&&&&&&\\ T&F&&&&&&\\ F&T&&&&&&\\ F&F&&&&&& \end{array} $$

  1. Use your truth table to prove that $\sim (P\vee Q)$ and $(\sim P)\wedge (\sim Q)$ are logically equivalent.
  2. Construct a similar truth table to prove that $\sim (P\wedge Q)$ and $(\sim P)\vee (\sim Q)$ are logically equivalent .

In other words, when you have finished this problem you will have shown that

the negation of a disjunction is the conjunction of the negations, and
the negation of a conjunction is the disjunction of the negations.

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