If the Boolean expression \((p\oplus q)\wedge(\sim p\odot q)\) is equivalent to \(p\wedge q\), where \(\oplus,\odot\in\{\wedge,\vee\}\), then the ordered pair \((\oplus,\odot)\) is :
Show Hint
- \((\vee,\wedge)\)
- \((\vee,\vee)\)
- \((\wedge,\vee)\)
- \((\wedge,\wedge)\)
The Correct Option is C
Solution and Explanation
(1) \((p\vee q)\wedge(\sim p\wedge q)=\sim p\wedge q\ne p\wedge q\).
(2) \((p\vee q)\wedge(\sim p\vee q)=q\vee(p\wedge\sim p)=q\ne p\wedge q\).
(3) \((p\wedge q)\wedge(\sim p\vee q)=p\wedge q\wedge(\sim p\vee q)\). When \(p\wedge q\) is true, \(q\) is true so \(\sim p\vee q\) is true; hence the expression equals \(p\wedge q\). Correct.
(4) \((p\wedge q)\wedge(\sim p\wedge q)=\) False (contains \(p\wedge\sim p\)).
Answer: (3) \((\wedge,\vee)\)
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