Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>Which of the following is logically equivalent to \(\sim(\sim p \rightarrow q)\)?</p>
<p>\(p \wedge q\)</p>
<p>\(p \wedge \sim q\)</p>
<p>\(\sim p \wedge q\)</p>
<p>\(\sim p \wedge \sim q\)</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's laws and the equivalence ∼(A → B) ≡ (A ∧ ∼B) to simplify the negation of a conditional statement. The key is recognizing that ∼(∼p → q) means the implication ∼p → q is false, which occurs exactly when ∼p is true AND q is false.
<p><strong>Step 1:</strong> Recall that the negation of a conditional statement: ∼(A → B) ≡ (A ∧ ∼B)</p><p><strong>Step 2:</strong> Apply this rule with A = ∼p and B = q:<br>∼(∼p → q) ≡ (∼p ∧ ∼q)</p><p><strong>Step 3:</strong> Simplify ∼p ∧ ∼q using De Morgan's law in reverse (or recognize it directly):<br>∼p ∧ ∼q ≡ ∼(p ∨ q)</p><p><strong>Step 4:</strong> Verify: ∼(∼p → q) is true when ∼p is true (p is false) AND q is false, which matches ∼(p ∨ q).</p><p>∴ <strong>Answer: ∼(p ∨ q) or equivalently (∼p ∧ ∼q)</strong></p>
Correct Answer: D