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

Step 1: The original expression We are given the statement \( \sim [p \vee (\sim (p \land q))] \). We need to simplify this expression and find the equivalent logical statement. 
Step 2: Apply De Morgan's law First, apply De Morgan’s law to the negation of the disjunction \( \sim [p \vee (\sim (p \land q))] \). De Morgan’s law states that \( \sim (A \vee B) = \sim A \land \sim B \), so we get: \[ \sim p \land \sim (\sim (p \land q)) \] 
Step 3: Simplify the inner negation Now, simplify the double negation \( \sim (\sim (p \land q)) \), which cancels out the two negations, giving us: \[ \sim p \land (p \land q) \] 
Step 4: Conclusion Thus, the expression simplifies to: \[ (p \land q) \land (\sim p) \] which is the correct equivalent form of the original expression.