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Which one of the following is a tautology?

JEE · Math · previous-year question

  1. A.P $$ \wedge $$ (P $$ \vee $$ Q)
  2. B.P $$ \vee $$ (P $$ \wedge $$ Q)
  3. C.Q $$ \to $$ (P $$ \wedge $$ (P $$ \to $$ Q))
  4. D.(P $$ \wedge $$ (P $$ \to $$ Q)) $$ \to $$ Qcorrect

Answer

D. (P $$ \wedge $$ (P $$ \to $$ Q)) $$ \to $$ Q

Explanation

Option A : .tg {border-collapse:collapse;border-spacing:0;} .tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg th{font-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg .tg-baqh{text-align:center;vertical-align:top} .tg .tg-wa1i{font-weight:bold;text-align:center;vertical-align:middle} .tg .tg-amwm{font-weight:bold;text-align:center;vertical-align:top} P Q P $$ \vee $$ Q P $$ \wedge $$ (P $$ \vee $$ Q) T T T T T F T T F T T F F F F F Option B : .tg {border-collapse:collapse;border-spacing:0;} .tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg th{font-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg .tg-baqh{text-align:center;vertical-align:top} .tg .tg-wa1i{font-weight:bold;text-align:center;vertical-align:middle} .tg .tg-amwm{font-weight:bold;text-align:center;vertical-align:top} P Q P $$ \wedge $$ Q P $$ \vee $$ (P $$ \wedge $$ Q) T T T T T F F T F T F F F F F F Option C : Q $$ \to $$ (P $$ \wedge $$ (P $$ \to $$ Q)) $$ \equiv $$ Q $$ \vee $$ (P $$ \wedge $$ (P $$ \to $$ Q)) $$ \equiv $$ Q $$ \vee $$ (P $$ \wedge $$ ( P $$ \vee $$ Q)) $$ \equiv $$ Q $$ \vee $$ (P $$ \wedge $$ P) $$ \vee $$ (P $$ \wedge $$ Q)) $$ \equiv $$ Q $$ \vee $$ (ƒ $$ \vee $$ (P $$ \wedge $$ Q)) $$ \equiv $$ Q $$ \vee $$ (P $$ \wedge$$ Q) $$ \equiv $$ ( Q $$ \vee $$ P) $$ \wedge $$ ( Q $$ \vee $$ Q) $$ \equiv $$ ( Q $$ \vee $$ P) $$ \wedge $$ t $$ \equiv $$ Q $$ \vee $$ P Option D : (P $$ \wedge $$ (P $$ \to $$ Q)) $$ \to $$ Q $$ \equiv $$ (P $$ \wedge $$ ( P $$ \vee $$ Q)) $$ \to $$ Q $$ \equiv $$ ((P $$ \wedge $$ P) $$ \vee $$ (P $$ \wedge $$ Q)) $$ \to $$ Q $$ \equiv $$ (ƒ $$ \vee $$ (P $$ \wedge $$ Q)) $$ \to $$ Q $$ \equiv $$ P $$ \wedge $$ Q $$ \to $$ Q $$ \equiv $$ (P $$ \wedge $$ Q) $$ \vee $$ Q $$ \equiv $$ ( P) $$ \vee $$ ( Q) $$ \vee $$ Q $$ \equiv $$ ( P) $$ \vee $$ t $$ \equiv $$ t

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