Question:
The dual of the statement pattern \((p\wedge q)\wedge \sim (p\vee q)\) is a:
The dual of the statement pattern \((p\wedge q)\wedge \sim (p\vee q)\) is a:
Show Hint
Interchange the conjunction and disjunction signs and then examine the truth values of the new pattern.
Updated On: Nov 12, 2026
- tautology
- contradiction
- logically equivalent to \(\sim p\wedge q\)
- logically equivalent to \(q\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
The dual of a statement pattern is found by replacing \(\wedge\) with \(\vee\) and \(\vee\) with \(\wedge\).
Step 2: Find the dual
The pattern \((p\wedge q)\wedge\sim(p\vee q)\) has dual \((p\vee q)\vee\sim(p\wedge q)\).
Step 3: Analyse the dual
If \(p\wedge q\) is true, then \(p\vee q\) is true.
If \(p\wedge q\) is false, then \(\sim(p\wedge q)\) is true.
Step 4: Conclusion
In every case at least one part is true, so the dual is always true.
Final Answer:
The dual is a tautology. \[ \boxed{\text{(A) }\text{tautology}} \]
The dual of a statement pattern is found by replacing \(\wedge\) with \(\vee\) and \(\vee\) with \(\wedge\).
Step 2: Find the dual
The pattern \((p\wedge q)\wedge\sim(p\vee q)\) has dual \((p\vee q)\vee\sim(p\wedge q)\).
Step 3: Analyse the dual
If \(p\wedge q\) is true, then \(p\vee q\) is true.
If \(p\wedge q\) is false, then \(\sim(p\wedge q)\) is true.
Step 4: Conclusion
In every case at least one part is true, so the dual is always true.
Final Answer:
The dual is a tautology. \[ \boxed{\text{(A) }\text{tautology}} \]
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