Mathematical Reasoning
Proposition Logic and Truth Tables
Grade 11
Question:
<p>The proposition <span>\(p \rightarrow \neg(p \wedge \neg q)\)</span> is equivalent to</p>
<p>(a) <span>\(q\)</span></p>
<p>(b) <span>\((\neg p) \vee q\)</span></p>
<p>(c) <span>\((\neg p) \wedge q\)</span></p>
<p>(d) <span>\((\neg p) \vee (\neg q)\)</span></p>
Step-by-Step Solution
Key Concept: Apply De-Morgan's Law to simplify the negation of a conjunction, then use the equivalence of implication to disjunction.
<p><strong>Solution:</strong> Given proposition <span>\(p \rightarrow \neg(p \wedge (\neg q))\)</span></p><p>By De-Morgan's Law: <span>\(= p \rightarrow (\neg p \vee q)\)</span></p><p><span>\(= (\neg p) \vee ((\neg p) \vee q)\)</span></p><p><span>\(= (\neg p) \vee q\)</span></p><p>Hence, option (b) is correct.</p>
Correct Answer: B