Sets, Relations & Functions
Mathematical Logic - Boolean Expressions
Grade 11
Question:
<p>The Boolean expression \(\sim(p \vee q) \vee (\sim p \vee q)\) is equivalent to</p>
<p>\(\sim q\)</p>
<p>\(\sim p\)</p>
<p>\(p\)</p>
<p>\(q\)</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's laws to simplify the negation, then use the associative and absorption properties of disjunction to reduce the expression to its simplest form.
<p><strong>Step 1:</strong> Apply De Morgan's law to ∼(p ∨ q)</p><p>∼(p ∨ q) = ∼p ∧ ∼q</p><p><strong>Step 2:</strong> Substitute into the original expression</p><p>(∼p ∧ ∼q) ∨ (∼p ∨ q)</p><p><strong>Step 3:</strong> Factor out ∼p using absorption</p><p>Since (A ∧ B) ∨ (A ∨ C) = A ∨ (B ∧ C) is not directly applicable, use the distributive approach:</p><p>= ∼p ∨ (∼p ∧ ∼q) ∨ q</p><p><strong>Step 4:</strong> Apply absorption: A ∨ (A ∧ B) = A</p><p>∼p ∨ (∼p ∧ ∼q) = ∼p</p><p><strong>Step 5:</strong> Final simplification</p><p>∼p ∨ q</p><p>∴ The expression is equivalent to <strong>∼p ∨ q</strong></p>
Correct Answer: B