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JEE Main 20236 AprilMorning Shift
Exam
JEE Main
Year
2023
Date
6 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Mathematical Reasoning

JEE Main 2023 · 6 April · Morning Shift · Mathematics · Mathematical Reasoning

Statement ( P ⇒ Q ) ∧ ( R ⇒ Q ) is logically equivalent to

Statement (P⇒Q)∧(R⇒Q)( P ⇒ Q ) ∧ ( R ⇒ Q ) is logically equivalent to

Options

  1. A(P⇒R)∨(Q⇒R)\left(P ⇒ R\right) ∨ \left(Q ⇒ R\right)
  2. B(P∧R)⇒Q\left(P ∧ R\right) ⇒ Q
  3. C(P⇒R)∧(Q⇒R)\left(P ⇒ R\right) ∧ \left(Q ⇒ R\right)
  4. D(P∨R)⇒Q\left(P ∨ R\right) ⇒ Q

Correct answer

D. (P∨R)⇒Q\left(P ∨ R\right) ⇒ Q

Step-by-step solution

Given that (P⇒Q)∧(R⇒Q)( P ⇒ Q ) ∧ ( R ⇒ Q )

We know that P⇒Q≡ P∨QP ⇒ Q ≡ ~ P ∨ Q

⇒( P∨Q)∧( R∨Q)⇒ \left(~ P ∨ Q\right) ∧ \left(~ R ∨ Q\right)

⇒( P∧ R)∨Q⇒ \left(~ P ∧ ~ R\right) ∨ Q

⇒ (P∨R)∨Q⇒ ~ \left(P ∨ R\right) ∨ Q

⇒(P∨R)⇒Q⇒ \left(P ∨ R\right) ⇒ Q

Hence this is the correct option.

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