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GATE | Gate IT 2005 | Question 36
• Difficulty Level : Hard
• Last Updated : 23 Dec, 2015

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)

Explanation:

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.

Quiz of this Question

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