Question:
The logically equivalent statement of \((\sim \mathrm{p} \wedge \mathrm{q}) \vee(\sim \mathrm{p} \wedge \sim \mathrm{q}) \vee(\mathrm{p} \wedge \sim \mathrm{q})\) is
The logically equivalent statement of \((\sim \mathrm{p} \wedge \mathrm{q}) \vee(\sim \mathrm{p} \wedge \sim \mathrm{q}) \vee(\mathrm{p} \wedge \sim \mathrm{q})\) is
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Group the first two terms using the common factor ~p, then combine with the third term.
Updated On: Nov 13, 2026
- \((\sim p)\wedge q\)
- \((\sim \mathrm{p})\vee (\sim \mathrm{q})\)
- \((\sim p)\wedge (\sim q)\)
- \(\mathrm{p}\vee \mathrm{q}\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
Use the distributive law and the fact that a statement or its negation is always true.
Step 2: Combine the first two terms
\((\sim p\wedge q)\vee(\sim p\wedge\sim q)=\sim p\wedge(q\vee\sim q)=\sim p\).
Step 3: Combine with the third term
\(\sim p\vee(p\wedge\sim q)=(\sim p\vee p)\wedge(\sim p\vee\sim q)=\sim p\vee\sim q\).
Step 4: Result
The statement is logically equivalent to \((\sim p)\vee(\sim q)\).
Final Answer:
The statement is equivalent to (~p) or (~q). \[ \boxed{\text{(B) }(\sim p)\vee(\sim q)} \]
Use the distributive law and the fact that a statement or its negation is always true.
Step 2: Combine the first two terms
\((\sim p\wedge q)\vee(\sim p\wedge\sim q)=\sim p\wedge(q\vee\sim q)=\sim p\).
Step 3: Combine with the third term
\(\sim p\vee(p\wedge\sim q)=(\sim p\vee p)\wedge(\sim p\vee\sim q)=\sim p\vee\sim q\).
Step 4: Result
The statement is logically equivalent to \((\sim p)\vee(\sim q)\).
Final Answer:
The statement is equivalent to (~p) or (~q). \[ \boxed{\text{(B) }(\sim p)\vee(\sim q)} \]
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