Sets, Relations & Functions
Mathematical Reasoning - Logical Equivalence
Grade 11
Question:
<p>The statement \(p \rightarrow (q \rightarrow p)\) is equivalent to</p>
<p>\(p \rightarrow (p \rightarrow q)\)</p>
<p>\(p \rightarrow (p \vee q)\)</p>
<p>\(p \rightarrow (p \wedge q)\)</p>
<p>\(p \rightarrow (p \leftrightarrow q)\)</p>
Step-by-Step Solution
Key Concept: Use the equivalence p → q ≡ ¬p ∨ q to convert the implication into disjunctive form, then apply De Morgan's laws and associativity to simplify.
<p><strong>Step 1:</strong> Convert the inner implication using p → q ≡ ¬q ∨ p</p><p>q → p ≡ ¬q ∨ p</p><p><strong>Step 2:</strong> Substitute back into the outer implication</p><p>p → (q → p) ≡ p → (¬q ∨ p)</p><p><strong>Step 3:</strong> Convert the outer implication using p → q ≡ ¬p ∨ q</p><p>p → (¬q ∨ p) ≡ ¬p ∨ (¬q ∨ p)</p><p><strong>Step 4:</strong> Apply associativity of disjunction</p><p>¬p ∨ (¬q ∨ p) ≡ (¬p ∨ p) ∨ ¬q</p><p><strong>Step 5:</strong> Recognize that ¬p ∨ p is a tautology (always true), so the entire expression simplifies</p><p>(¬p ∨ p) ∨ ¬q ≡ T ∨ ¬q ≡ <strong>T (Tautology)</strong></p><p>∴ Answer: B (The statement is a tautology/always true)</p>
Correct Answer: B