Mathematical Reasoning
Boolean Algebra and Logic
Grade 11
Question:
<p>The negation of the boolean expression <span>\((p \rightarrow (q \rightarrow p)) \rightarrow \neg s \vee (\neg r \wedge s)\)</span> is equivalent to</p>
<p>(a) <span>\(s \wedge r\)</span></p>
<p>(b) <span>\(\neg s \wedge \neg r\)</span></p>
<p>(c) <span>\(s \vee r\)</span></p>
<p>(d) <span>\(r\)</span></p>
Step-by-Step Solution
Key Concept: Apply De-Morgan's Laws systematically to negate a complex boolean expression, then simplify using distributive laws.
<p><strong>Solution:</strong> The given boolean expression is <span>\(\neg s \vee ((\neg r) \wedge s)\)</span></p><p>The negation of the given boolean expression is:</p><p><span>\(\neg(\neg s \vee ((\neg r) \wedge s))\)</span></p><p><span>\(= s \wedge \neg((\neg r) \wedge s)\)</span> [By De-Morgan's Law: <span>\(\neg(p \vee q) = \neg p \wedge \neg q\)</span>]</p><p><span>\(= s \wedge (r \vee (\neg s))\)</span> [By De-Morgan's Law: <span>\(\neg(p \wedge q) = \neg p \vee \neg q\)</span>]</p><p><span>\(= (s \wedge r) \vee (s \wedge (\neg s))\)</span> [By Distributive Law: <span>\(p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)\)</span>]</p><p><span>\(= (s \wedge r)\)</span></p><p>Hence, option (a) is correct.</p>
Correct Answer: A