Question

The statement ~[p v (~(p ∧ q))] is equivalent to

• A. (~(р ∧ q)) ∧ q
• B. ~ (р ∧ q)
• C. (р ∧ q) ∧ (~р)
• D. ~(p v q)

Answer 1

The Correct Option is C

To determine the statement equivalent to ~[p \lor (~(p \land q))], we will apply logical equivalences step-by-step:

1. Start with the given expression: ~[p \lor (~(p \land q))].
2. Apply De Morgan's laws: \lnot (A \lor B) = (\lnot A) \land (\lnot B).
   • The expression is transformed to: (\lnot p) \land (p \land q).
3. Simplify the expression:
   • Distribute \lnot p inside the parenthesis: (\lnot p \land p) \land (\lnot p \land q).
4. Observe the term (\lnot p \land p), commonly known as a contradiction, which typically returns false.
   • In Boolean logic, a contradiction results in zero, making the entire expression evaluate to zero in conventional terms, yet in symbolic representation remains as: (\lnot p \land q).
5. The equivalent expression matches option (c): (p \land q) \land (\lnot p), which we now see aligns with the derived expression:
6. The correct answer is Option (c): (p \land q) \land (\lnot p).

Thus, using logical equivalences and transformations, we can conclude that option (c) is the correct equivalent.