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Prove that (p q) is logically equivalent to p q using a truth table.

Computer Science

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Step-by-step solution

3 steps
  1. Construct a truth table for the expressions ¬(p∨q)\neg (p \lor q) and ¬p∧¬q\neg p \land \neg q.

    We need columns for pp, qq, p∨qp \lor q, ¬(p∨q)\neg (p \lor q), ¬p\neg p, ¬q\neg q, and ¬p∧¬q\neg p \land \neg q.

  2. Fill in the truth values for all possible combinations of pp and qq.

    pqp∨q¬(p∨q)¬p¬q¬p∧¬qTTTFFFFTFTFFTFFTTFTFFFFFTTTT\begin{array}{|c|c|c|c|c|c|c|} \hline p & q & p \lor q & \neg (p \lor q) & \neg p & \neg q & \neg p \land \neg q \\ \hline T & T & T & F & F & F & F \\ T & F & T & F & F & T & F \\ F & T & T & F & T & F & F \\ F & F & F & T & T & T & T \\ \hline \end{array}
  3. Compare the columns for ¬(p∨q)\neg (p \lor q) and ¬p∧¬q\neg p \land \neg q.

    The column for ¬(p∨q)\neg (p \lor q) is F,F,F,TF, F, F, T.
    The column for ¬p∧¬q\neg p \land \neg q is F,F,F,TF, F, F, T.
    Since the truth values in these two columns are identical for all possible combinations of pp and qq, the expressions are logically equivalent.

    Therefore, ¬(p∨q)\neg (p \lor q) and ¬p∧¬q\neg p \land \neg q are logically equivalent.

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