GATE | Gate IT 2008 | Question 21

Which of the following is the negation of [∀ x, α → (∃y, β → (∀ u, ∃v, y))]
(A) [∃ x, α → (∀y, β → (∃u, ∀ v, y))]
(B) [∃ x, α → (∀y, β → (∃u, ∀ v, ¬y))]
(C) [∀ x, ¬α → (∃y, ¬β → (∀u, ∃ v, ¬y))]
(D) [∃ x, α ʌ (∀y, β ʌ (∃u, ∀ v, ¬y))]

Explanation:

When we negate a quantified statement, we negate all the quantifiers first, from left to right (keeping the same order), then we negative the statement.

We can take examples as:
1. ¬[∀x ∈ A, P(x)] ⇔ ∃x ∈ A, ¬P(x).
2. ¬[∃x ∈ A, P(x)] ⇔ ∀x ∈ A, ¬P(x).
3. ¬[∀x ∈ A, ∃y ∈ B, P(x, y)] ⇔ ∃x ∈ A, ∀y ∈ B, ¬P(x, y).
4. ¬[∃x ∈ A, ∀y ∈ B, P(x, y)] ⇔ ∀x ∈ A, ∃y ∈ B, ¬P(x, y).

important, to negate an implication:
¬[IF P, THEN Q] ⇔ P AND NOT Q

Now question is to negate this qualified statement:
[∀ x, a -> (∃y, B -> (∀u, ∃v,y))]

By negating it:
¬ [∀ x, a -> (∃y, B -> (∀u, ∃v,y))]
x, a ^ { ¬(∃y) , ¬[ B -> (∀u, ∃v,y) ] }
x, a ^ ( ∀y, B ^ ¬ (∀u, ∃v,y))
x, a ^ ( ∀y, B ^ (∃u, ∀v, ¬y))

option D is correct.

This solution is contributed by Nitika Bansal .

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