Question

Find the equivalent of \( p \land (q \lor r) \lor \neg (r \land \neg (p \land q)) \).

Answer 1

We are tasked with simplifying the logical expression: \[ p \land (q \lor r) \lor \neg (r \land \neg (p \land q)) \]

Step 1: Apply Distribution: First, we apply the distributive property to \( p \land (q \lor r) \). Using the distributive property of conjunction over disjunction, we get: \[ p \land (q \lor r) = (p \land q) \lor (p \land r) \] So, the expression becomes: \[ (p \land q) \lor (p \land r) \lor \neg (r \land \neg (p \land q)) \]

Step 2: Simplify the Negation: Now, simplify the negation \( \neg (r \land \neg (p \land q)) \). By applying De Morgan's law to the negation, we get: \[ \neg (r \land \neg (p \land q)) = \neg r \lor \neg (\neg (p \land q)) = \neg r \lor (p \land q) \] So the expression becomes: \[ (p \land q) \lor (p \land r) \lor (\neg r \lor (p \land q)) \]

Step 3: Combine Like Terms: Notice that \( (p \land q) \) appears twice. So we can eliminate the redundancy: \[ (p \land q) \lor (p \land r) \lor \neg r \] Now, we can group terms to simplify further: \[ (p \land q) \lor (\neg r \land p) \lor \neg r \]

Step 4: Final Simplification: Finally, we can simplify further by observing that the expression is equivalent to: \[ p \lor (\neg r \land q) \] This is the simplified version of the given logical expression.

Final Answer: Therefore, the equivalent of the given expression is: \[ p \lor (\neg r \land q) \]