Sets, Relations & Functions
Mathematical Logic - Negation
Grade 11
Question:
<p>The negation of \(\sim s \vee (\sim r \wedge s)\) is equivalent to</p>
<p>\(s \wedge \sim r\)</p>
<p>\(s \wedge (r \wedge \sim s)\)</p>
<p>\(s \vee (r \vee \sim s)\)</p>
<p>\(s \wedge r\)</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's Laws systematically: the negation of a disjunction becomes a conjunction of negations, and negate each component separately including compound expressions.
<p><strong>Step 1:</strong> Identify the structure of the expression: ∼s ∨ (∼r ∧ s)</p><p>This is a disjunction (∨) of two parts: [∼s] and [∼r ∧ s]</p><p><strong>Step 2:</strong> Apply De Morgan's Law to the negation of the disjunction:</p><p>∼[∼s ∨ (∼r ∧ s)] = ∼(∼s) ∧ ∼(∼r ∧ s)</p><p><strong>Step 3:</strong> Simplify ∼(∼s):</p><p>∼(∼s) = s</p><p><strong>Step 4:</strong> Apply De Morgan's Law to ∼(∼r ∧ s):</p><p>∼(∼r ∧ s) = ∼(∼r) ∨ ∼s = r ∨ ∼s</p><p><strong>Step 5:</strong> Combine the results from Steps 3 and 4:</p><p>s ∧ (r ∨ ∼s)</p><p><strong>Step 6:</strong> Optionally expand using distributive law if needed:</p><p>s ∧ (r ∨ ∼s) = (s ∧ r) ∨ (s ∧ ∼s) = (s ∧ r) ∨ F = s ∧ r</p><p>∴ Answer: D (which is s ∧ r or its equivalent form)</p>
Correct Answer: D