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# GATE | GATE-IT-2004 | Question 3

• Last Updated : 07 Apr, 2021
Let a(x, y), b(x, y,) and c(x, y) be three statements with variables x and y chosen from some universe. Consider the following statement:

(∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)]

Which one of the following is its equivalent?

(A) (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)]
(B) (∃x)(∀y)[(a(x, y) ∨ b(x, y)) ∧¬ c(x, y)]
(C) ¬ (∀x)(∃y)[(a(x, y) ∧ b(x, y)) → c(x, y)]
(D) ¬ (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)]

Answer: (C)

Explanation: Choice (C) is

= ¬ (∀x)(∃y)[(a(x, y) ∧ b(x, y)) → c(x, y)]

= ¬ (∀x)(∃y)[ a∧ b → c ]

= ¬ (∀x)(∃y)[ (ab)’ + c]

= ∃ x ∀ y[ (ab)’ + c]’

= ∃ x ∀ y[ abc’ ]

= ∃ x ∀ y[ a ∧ b ∧ c ¬c]

Which is same as the given expression.

(∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)]

Quiz of this Question

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