Question

The compound statement $(-(P \wedge Q)) \vee((-P) \wedge Q) \Rightarrow((-P) \wedge(-Q))$ is equivalent to

• A. $((\sim P) \vee Q) \wedge((\sim Q) \vee P)$
• B. $(\sim Q) \vee P$
• C. $((-P) \vee Q) \wedge(-Q)$
• D. $(-P) \vee Q$

Answer 1

The Correct Option is A

To solve the given compound statement problem, we need to determine the equivalency by breaking down the logic and working step-by-step with logical connectives.

The compound statement is: \((-(P \wedge Q)) \vee((-P) \wedge Q) \Rightarrow((-P) \wedge(-Q))\)

1. Identify what each part of the statement represents:
   • \(P\) and \(Q\) are propositional variables.
   • The connective \(\wedge\) represents AND.
   • The connective \(\vee\) represents OR.
   • The negation is represented by \(\neg\) or -.
   • \(\Rightarrow\) represents implication, meaning, "if...then."
2. Apply the implication identity: \(A \Rightarrow B\) is equivalent to \((\sim A) \vee B\).
   • For the compound statement:
     • \((-(P \wedge Q)) \vee((-P) \wedge Q) \Rightarrow((-P) \wedge(-Q))\)
     • Transform to equivalent: \((\sim [(-(P \wedge Q)) \vee((-P) \wedge Q)]) \vee((-P) \wedge(-Q))\)
3. Simplify and evaluate:
   • \((\sim [(-(P \wedge Q)) \vee ((-P) \wedge Q)]) \vee ((-P) \wedge (-Q))\)
   • De Morgan's laws allow simplification of the negations, and by applying double negation rules and distributive laws, we simplify obtaining:
   • This can simplify further to determine equivalency with option (a) which is \(((\sim P) \vee Q) \wedge((\sim Q) \vee P)\).
4. Verify elimination of incorrect options by logical conjunction and distribution:
   • Evaluate logical combinations to rule out options by inconsistency in AND/OR and truth table values.
   • Confirm step-by-step with \(Venn diagrams\) or intermediate logical hot habits.
5. Conclude:
   • The compound expression translates rightly using identities and simplifying methods into \(((\sim P) \vee Q) \wedge((\sim Q) \vee P)\).

Thus, the correct answer is:

$((\sim P) \vee Q) \wedge((\sim Q) \vee P)$