Verified Solution • 2025–26 Session moderate

QJEEM20Ma3efd: JEE Main 2020 Mathematics - Class 11 JEEM20Ma3efd Solution: Which one of the following is a tautology?

Which one of the following is a tautology?

Multiple Choice Options

  • A P ∧\wedge (P ∨\vee Q)
  • B P ∨\vee (P ∧\wedge Q)
  • C Q →\to (P ∧\wedge (P →\to Q))
  • D (P ∧\wedge (P →\to Q)) →\to Q

Step-by-Step Verified Solution

Option A :

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 :

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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