Sets, Relations & Functions
Mathematical Logic
Grade 11
Question:
<p>The logically equivalent statement of \(p \Rightarrow (q \lor r)\) is:</p>
<p>(1) \((p \Rightarrow q) \land (p \Rightarrow r)\)</p>
<p>(2) \((p \Rightarrow q) \lor (p \Rightarrow r)\)</p>
<p>(3) \((p \lor q) \Rightarrow r\)</p>
<p>(4) \((p \Rightarrow q) \Rightarrow r\)</p>
Step-by-Step Solution
Key Concept: Use the equivalence p ⇒ q ≡ ¬p ∨ q, then apply De Morgan's laws to simplify the resulting logical expression into a recognizable form.
<p><strong>Step 1:</strong> Convert implication to disjunction using p ⇒ q ≡ ¬p ∨ q</p><p>p ⇒ (q ∨ r) ≡ ¬p ∨ (q ∨ r)</p><p><strong>Step 2:</strong> Apply associativity of disjunction</p><p>¬p ∨ (q ∨ r) ≡ (¬p ∨ q) ∨ r</p><p><strong>Step 3:</strong> Recognize this can be rewritten as (p ⇒ q) ∨ r or ¬(p ∧ ¬q ∧ ¬r)</p><p>The logically equivalent form is: <strong>¬p ∨ q ∨ r</strong> or equivalently <strong>(p ∧ ¬q) ⇒ r</strong></p><p>∴ Answer: B</p>
Correct Answer: B