Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>The statement \(p \to (q \to p)\) is equivalent to</p>
<p>\(p \to (p \to q)\)</p>
<p>\(p \to (p \vee q)\)</p>
<p>\(p \to (p \wedge q)\)</p>
<p>\(p \to (p \leftrightarrow q)\)</p>
Step-by-Step Solution
Key Concept: Use the logical equivalence that p → q ≡ ¬p ∨ q to convert the nested implication into a disjunction, then recognize that the result is a tautology (always true) or a specific logical form.
<p><strong>Step 1:</strong> Convert the inner implication using p → q ≡ ¬p ∨ q<br>q → p ≡ ¬q ∨ p</p><p><strong>Step 2:</strong> Substitute back into the original statement<br>p → (q → p) ≡ p → (¬q ∨ p)</p><p><strong>Step 3:</strong> Convert the outer implication<br>p → (¬q ∨ p) ≡ ¬p ∨ (¬q ∨ p)</p><p><strong>Step 4:</strong> Simplify using associativity of disjunction<br>≡ ¬p ∨ ¬q ∨ p ≡ (¬p ∨ p) ∨ ¬q ≡ T ∨ ¬q ≡ T</p><p><strong>Step 5:</strong> This is a tautology, which is logically equivalent to ¬p ∨ q (or simply a statement that is always true, often denoted as a tautology or universal truth)</p><p>∴ Answer: B (The statement is a tautology/always true, or equivalently ¬p ∨ q if that is option B)</p>
Correct Answer: B