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Problem 18: (The Negation Of An Implication Is A Conjunction)

In this problem we want to find the negation of $P\implies Q$.

  1. In a truth table for an implication $P\implies Q$, how many of the 4 rows contain the truth value $T$?
  2. In a truth table for a conjuction $P\wedge Q$, how many of the 4 rows contain the truth value $T$?
  3. In a truth table for a disjunction $P\vee Q$, how many of the 4 rows contain the truth value $T$?
  4. In a truth table for the negation of the implication $P\implies Q$, how many of the 4 rows contain the truth value $T$? Based off this answer alone, explain why you expect the negation of an implication to be a conjunction.
  5. Complete the truth table below, and use your answer to determine which statement is logically equivalent to $\sim(P\implies Q)$.

$$ \begin{array}{c|c|c|c|c|c|c|c} P&Q&P\implies Q&\sim(P\implies Q) &P \wedge Q &P \wedge (\sim Q) & (\sim P) \wedge Q& (\sim P) \wedge (\sim Q) \\\hline T&T&&&&&&\\ T&F&&&&&&\\ F&T&&&&&&\\ F&F&&&&&& \end{array} $$

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