The statement ~[p v (~(p ∧ q))] is equivalent to
- (~(р ∧ q)) ∧ q
- ~ (р ∧ q)
- (р ∧ q) ∧ (~р)
- ~(p v q)
The Correct Option is C
Solution and Explanation
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.
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