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Step-by-step solution

7 steps

Here are the steps to construct the truth table for the given logical expression:

[(P→Q)∧(¬P∨¬Q)]↔P[(P \to Q) \land (\neg P \lor \neg Q)] \leftrightarrow P
  1. List all possible truth values for PP and QQ.

  2. Calculate the truth values for P→QP \to Q.

    This is false (F) only when PP is true (T) and QQ is false (F).

  3. Calculate the truth values for ¬P\neg P.

    This is the opposite of the values for PP.

  4. Calculate the truth values for ¬Q\neg Q.

    This is the opposite of the values for QQ.

  5. Calculate the truth values for ¬P∨¬Q\neg P \lor \neg Q.

    This is false (F) only when both ¬P\neg P and ¬Q\neg Q are false (F).

  6. Calculate the truth values for (P→Q)∧(¬P∨¬Q)(P \to Q) \land (\neg P \lor \neg Q).

    This is true (T) only when both (P→Q)(P \to Q) and (¬P∨¬Q)(\neg P \lor \neg Q) are true (T).

  7. Calculate the truth values for the final expression [(P→Q)∧(¬P∨¬Q)]↔P[(P \to Q) \land (\neg P \lor \neg Q)] \leftrightarrow P.

    This is true (T) only when both sides, i.e., (P→Q)∧(¬P∨¬Q)(P \to Q) \land (\neg P \lor \neg Q) and PP, have the same truth value.

    The truth table is as follows:

    PQP→Q¬P¬Q¬P∨¬Q(P→Q)∧(¬P∨¬Q)[(P→Q)∧(¬P∨¬Q)]↔PTTTFFFFFTFFFTTFFFTTTFTTFFFTTTTTF\begin{array}{|c|c|c|c|c|c|c|c|} \hline P & Q & P \to Q & \neg P & \neg Q & \neg P \lor \neg Q & (P \to Q) \land (\neg P \lor \neg Q) & [(P \to Q) \land (\neg P \lor \neg Q)] \leftrightarrow P \\ \hline T & T & T & F & F & F & F & F \\ T & F & F & F & T & T & F & F \\ F & T & T & T & F & T & T & F \\ F & F & T & T & T & T & T & F \\ \hline \end{array}

    The final column of the truth table shows all 'F' (False) values. Therefore, the given expression is a contradiction.

    The final column of the truth table consists of all F values, indicating a contradiction.

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