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GATE | Gate IT 2005 | Question 36

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  • Difficulty Level : Hard
  • Last Updated : 28 Jun, 2021

Let P(x) and Q(x) be arbitrary predicates. Which of the following statements is always TRUE?
(A) ((∀x(P(x)∨Q(x))))⟹((∀xP(x))∨(∀xQ(x)))
(B) (∀x(P(x)⟹Q(x)))⟹((∀xP(x))⟹(∀xQ(x)))
(C) (∀x(P(x))⟹∀x(Q(x)))⟹(∀x(P(x)⟹Q(x)))
(D) (∀x(P(x))⇔(∀x(Q(x))))⟹(∀x(P(x)⇔Q(x)))

Answer: (B)


Generally, these type of questions can be solved by using two statements and checking the validity of each and every option.
Here, let the statements be P and Q where,
P : Student is a girl
Q : Student is smart

Option B says that, IF for all student x : If x is a girl then the student is smart THEN IF the whole class comprises of girls then the whole class comprises of smart students.

This solution is contributed by Anil Saikrishna Devarasetty.

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