Sets, Relations & Functions
Mathematical Reasoning
Grade None
Question:
<p>The inverse of the proposition \((p \wedge \sim q) \rightarrow r\) is</p>
<p>\(\sim r \rightarrow (\sim p \vee q)\)</p>
<p>\(r \rightarrow p \wedge \sim q\)</p>
<p>\((\sim p \vee q) \rightarrow \sim r\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: The inverse of a conditional proposition p → q is ~p → ~q. For a compound antecedent, apply negation to the entire antecedent and the consequent separately.
<p><strong>Step 1:</strong> Identify the structure of the given proposition: (p ∧ ~q) → r</p><p>Here, antecedent = (p ∧ ~q) and consequent = r</p><p><strong>Step 2:</strong> Apply the definition of inverse. If the original proposition is A → B, then the inverse is ~A → ~B</p><p><strong>Step 3:</strong> Negate the antecedent: ~(p ∧ ~q)</p><p>Using De Morgan's Law: ~(p ∧ ~q) = (~p ∨ ~~q) = (~p ∨ q)</p><p><strong>Step 4:</strong> Negate the consequent: ~r</p><p><strong>Step 5:</strong> Form the inverse: (~p ∨ q) → ~r</p><p>∴ Answer: C (which represents (~p ∨ q) → ~r)</p>
Correct Answer: C