Question:

<p><strong>Step 1:</strong> Start with $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$</p><p><strong>Step 2:</strong> Rearrange as $(p \wedge \sim q) \vee (q \vee (\sim p \wedge q))$</p><p><strong>Step 3:</strong> Apply absorption: $q \vee (\sim p \wedge q) = (q \vee \sim p) \wedge (q \vee q) = (q \vee \sim p) \wedge q = q$</p><p>Alternatively: $q \vee (\sim p \wedge q) = q$ (since if q is true, the whole disjunction is true; if q is false, then $(\sim p \wedge q)$ is false)</p><p><strong>Step 4:</strong> Expression becomes $(p \wedge \sim q) \vee q$</p><p><strong>Step 5:</strong> This is equivalent to $p \vee q$ (by absorption law: $(p \wedge \sim q) \vee q = (p \vee q) \wedge (\sim q \vee q) = (p \vee q) \wedge T = p \vee q$)</p><p><strong>Verification by truth table:</strong> When p=T,q=T: T; p=T,q=F: T; p=F,q=T: T; p=F,q=F: F</p><p>∴ Answer: $p \vee q$ (Option D)</p>