Question

The negation of the expression $q \vee((\sim q) \wedge p)$ is equivalent to

• A. $p \wedge(\sim q)$
• B. $(\sim p) \wedge(\sim q)$
• C. $(\sim p) \vee(\sim q)$
• D. $(\sim p) \vee q$

Answer 1

The Correct Option is B

 To solve the given problem, we need to find the negation of the logical expression \(q \vee((\sim q) \wedge p)\). Let's break it down step by step:

1. First, understand the original expression:
   • \(q\) represents a statement.
   • \(\sim q\) represents the negation of \(q\).
   • \(p\) represents another statement.
   • \sim q and \(p\).
   • Finally, \(q \vee((\sim q) \wedge p)\) is the disjunction ("or") between \(q\) and the conjunction \sim (q \vee r) = (\sim q) \wedge (\sim r).
   • Thus, \(\sim[q \vee((\sim q) \wedge p)] = (\sim q) \wedge \sim((\sim q) \wedge p)\).
2. Negate the conjunction \sim((\sim q) \wedge p) = (\sim (\sim q)) \vee (\sim p).
3. This simplifies to \(q \vee (\sim p)\).
4. Substitute back into the main expression:
   • \((\sim q) \wedge (q \vee (\sim p))\).
   • Distribute \((\sim q)\): \(((\sim q) \wedge q) \vee ((\sim q) \wedge (\sim p))\).
   • The term (\sim q) \wedge (\sim p).
5. Thus, the negation of the expression \(q \vee((\sim q) \wedge p)\) is equivalent to