Sets, Relations & Functions
Mathematical Reasoning / Boolean Expressions
Grade 11
Question:
<p>The Boolean expression <span>\(\sim(p \Rightarrow (\sim q))\)</span> is equivalent to</p>
<p>\(p \wedge q\)</p>
<p>\(q \Rightarrow \sim p\)</p>
<p>\(p \vee q\)</p>
<p>\((\sim p) \Rightarrow q\)</p>
Step-by-Step Solution
Key Concept: Negate the implication by converting ¬(p → r) to (p ∧ ¬r), then apply double negation to simplify ¬(¬q) back to q.
<p><strong>Step 1:</strong> Recall that the negation of an implication is: ¬(p → q) ≡ (p ∧ ¬q)</p><p><strong>Step 2:</strong> Apply this rule to ¬(p → (¬q)):<br/>¬(p → (¬q)) ≡ p ∧ ¬(¬q)</p><p><strong>Step 3:</strong> Simplify ¬(¬q) using double negation:<br/>p ∧ ¬(¬q) ≡ p ∧ q</p><p><strong>Step 4:</strong> Therefore, ¬(p → (¬q)) ≡ <strong>p ∧ q</strong></p><p>∴ Answer: A</p>
Correct Answer: A