Sets, Relations & Functions
Mathematical Reasoning / Negation
Grade 11
<p>Given \(\sim s \vee (\sim r \wedge s)\). The negation of this statement is:</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's laws systematically to negate a compound logical statement: negate the main operator first, then distribute negation inward, remembering that ¬(A ∨ B) = ¬A ∧ ¬B and ¬(A ∧ B) = ¬A ∨ ¬B.
<p><strong>Step 1:</strong> Identify the main structure. The statement is ∼s ∨ (∼r ∧ s), a disjunction.</p><p><strong>Step 2:</strong> Apply negation: ¬[∼s ∨ (∼r ∧ s)]</p><p><strong>Step 3:</strong> Use De Morgan's law ¬(A ∨ B) = ¬A ∧ ¬B:<br/>= ¬(∼s) ∧ ¬(∼r ∧ s)</p><p><strong>Step 4:</strong> Simplify ¬(∼s) = s, and apply De Morgan's law to ¬(∼r ∧ s) = ¬(∼r) ∨ ¬s = r ∨ ¬s:<br/>= s ∧ (r ∨ ∼s)</p><p><strong>Step 5:</strong> This can also be written as (s ∧ r) ∨ (s ∧ ∼s). Since (s ∧ ∼s) is always false, the result simplifies to s ∧ r.</p><p>∴ <strong>Answer: s ∧ r</strong> (or equivalently written as (s ∧ r) ∨ (s ∧ ∼s) in expanded form, depending on how option B is stated)</p>
Correct Answer: B