Sets, Relations & Functions
Mathematical Logic - Boolean Expressions
Grade None
<p>The Boolean Expression \((p \wedge \sim q) \vee q \vee (\sim p \wedge q)\) is equivalent to</p>
Step-by-Step Solution
Key Concept: Simplify the Boolean expression by grouping terms strategically and applying absorption laws: notice that (p ∧ ¬q) ∨ q can be simplified using the fact that (A ∨ q) absorbs q when combined appropriately, and recognize that ¬p ∧ q is already in disjunctive form with q.
<p><strong>Step 1:</strong> Rearrange the expression: (p ∧ ¬q) ∨ q ∨ (¬p ∧ q)</p><p><strong>Step 2:</strong> Simplify (p ∧ ¬q) ∨ q using the absorption law. Note that (p ∧ ¬q) ∨ q = (p ∨ q) ∧ (¬q ∨ q) = (p ∨ q) ∧ T = (p ∨ q)</p><p><strong>Step 3:</strong> Now we have (p ∨ q) ∨ (¬p ∧ q)</p><p><strong>Step 4:</strong> Since (p ∨ q) already contains q as a disjunct, and (¬p ∧ q) is subsumed by q in (p ∨ q), the expression simplifies to (p ∨ q)</p><p><strong>Verification:</strong> When q = T, the entire expression is T. When q = F, we get (p ∧ T) ∨ F ∨ (¬p ∧ F) = p, which matches p ∨ F = p. Thus (p ∨ q) is correct.</p><p>∴ Answer: <strong>p ∨ q</strong> (Option B)</p>
Correct Answer: B