Question

Using the rules in logic, write the negation of the following: \[ (p q) \land (q \lor \sim r) \]

Hint:

Use De Morgan's laws when negating conjunctions or disjunctions: \[ \sim (A \land B) = \sim A \lor \sim B \quad \text{and} \quad \sim (A \lor B) = \sim A \land \sim B. \]

Answer 1

To determine the negation of \( (p q) \land (q \lor \sim r) \), De Morgan's laws will be utilized. 1. Initiate by negating the complete expression:\[\sim \left( (p q) \land (q \lor \sim r) \right)\]Applying De Morgan's law for conjunctions states that the negation of a conjunction is the disjunction of the negations:\[\sim (p q) \lor \sim (q \lor \sim r).\]2. Subsequently, apply De Morgan’s law to both \( \sim (p q) \) and \( \sim (q \lor \sim r) \):- \( \sim (p q) \) transforms to \( \sim p \lor \sim q \)- \( \sim (q \lor \sim r) \) transforms to \( \sim q \land r \)Consequently, the negation of the initial expression is expressed as:\[(\sim p \lor \sim q) \lor (\sim q \land r).\]3. Further simplification yields the final result:\[\sim q \land (\sim p \lor r).\]