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 find the negation of \( (p q) \land (q \lor \sim r) \), we will apply De Morgan's laws. 1. First, apply the negation to the entire expression: \[ \sim \left( (p q) \land (q \lor \sim r) \right) \] According to De Morgan's law, the negation of a conjunction is the disjunction of the negations: \[ \sim (p q) \lor \sim (q \lor \sim r). \] 2. Now, apply De Morgan’s law to \( \sim (p q) \) and \( \sim (q \lor \sim r) \): - \( \sim (p q) = \sim p \lor \sim q \) - \( \sim (q \lor \sim r) = \sim q \land r \) Thus, the negation of the original expression becomes: \[ (\sim p \lor \sim q) \lor (\sim q \land r). \] 3. Simplifying this further, we get the final result: \[ \sim q \land (\sim p \lor r). \]