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The dual of (p ^^ t) vv ( c ^^ ~ q) wh...

The dual of ` (p ^^ t) vv ( c ^^ ~ q)` where t is a tautology and c is a contradiction, is

A

`(p vv c) ^^ (t vv ~ q)`

B

`(~p ^^ c) ^^ ( t vv q)`

C

`(~ p vv c) ^^ (t vv q)`

D

`(~p vv t) ^^ ( c vv ~ q)`

Text Solution

AI Generated Solution

The correct Answer is:
To find the dual of the expression \( (p \land t) \lor (c \land \neg q) \), where \( t \) is a tautology and \( c \) is a contradiction, we will follow a systematic approach. ### Step-by-Step Solution: 1. **Identify the components of the expression:** - The expression is \( (p \land t) \lor (c \land \neg q) \). - Here, \( t \) is a tautology (always true) and \( c \) is a contradiction (always false). 2. **Apply the duality principle:** - In duality, we change: - Conjunction (\( \land \)) to disjunction (\( \lor \)) - Disjunction (\( \lor \)) to conjunction (\( \land \)) - Tautology (\( t \)) to contradiction (\( c \)) - Contradiction (\( c \)) to tautology (\( t \)) 3. **Change the conjunctions and disjunctions:** - The first part \( (p \land t) \) becomes \( (p \lor c) \) because \( \land \) changes to \( \lor \) and \( t \) changes to \( c \). - The second part \( (c \land \neg q) \) becomes \( (t \lor \neg q) \) because \( \land \) changes to \( \lor \) and \( c \) changes to \( t \). 4. **Combine the transformed parts:** - Now we combine the transformed parts: \[ (p \lor c) \land (t \lor \neg q) \] 5. **Final expression:** - The dual of the original expression \( (p \land t) \lor (c \land \neg q) \) is: \[ (p \lor c) \land (t \lor \neg q) \] ### Final Answer: The dual of \( (p \land t) \lor (c \land \neg q) \) is \( (p \lor c) \land (t \lor \neg q) \).
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TARGET PUBLICATION-MATHEMATICAL LOGIC -CRITICAL THINKING
  1. Which of the following is logically equivalent to ~[p to (p vv ~ q)] ...

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  2. Which of the following is true?

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  3. ~ ( ~p) harr p is

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  4. (~ p ^^ ~ q) ^^ ( q ^^ r) is a

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  5. Which of the following is a tautology?

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  6. Which of the following statement is contradiction?

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  7. Which of the following statement is a contingency?

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  8. If A -= {4, 5, 7, 9), determine which of the following quantified stat...

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  9. Using quantifier the open sentence x^2 gt 0 ' defined on N is converte...

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  10. Which of the following quantified statement is false?

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  11. Given below are four statements along with their respective duals. Whi...

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  12. The dual of (p ^^ t) vv ( c ^^ ~ q) where t is a tautology and c is ...

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  13. Negation of the proposition (p vv q) ^^(~q^^r) is

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  14. The negation of p vv ~ q is

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  15. Which of the following is logically equivalent to ~[ p to ( p vv ~q)]...

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  16. The negation of the statement, EE x in R such that x^2 + 3 gt 0 , is

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  17. The negation of the statement "If Saral Mart does not reduce the price...

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  18. The switching circuit for the statement [p ^^ (q vv r) ] vv (~p vv s) ...

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  19. If the symbolic form is (p ^^ r) vv (~q ^^ ~r) vv (~q ^^ ~r), then sw...

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  20. The switching circuit for the symbolic form (p vv q) ^^ [~p vv (r ^^ ~...

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