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Which of the following is the negation of $[∀ x, α → (∃y, β → (∀ u, ∃v, y))]$

  1. $[∃ x, α → (∀y, β → (∃u, ∀ v, y))]$
  2. $[∃ x, α → (∀y, β → (∃u, ∀ v, ¬y))]$
  3. $[∀ x, ¬α → (∃y, ¬β → (∀u, ∃ v, ¬y))]$
  4. $[∃ x, α \wedge (∀y, β \wedge (∃u, ∀ v, ¬y))]$
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Best answer
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$[∀ x, α → (∃y, β → (∀ u, ∃v, y))] \equiv [∀ x, ¬α \vee  (∃y, ¬β v (∀ u, ∃v, y))]$

Now, doing complement gives (complement of $∀$ is $∃$ and vice versa while propagating negation inwards as $∀x (P) \equiv ¬∃x (¬P)$ and $∃x (P) \equiv ¬∀x (¬P))$

$[∃ x, α \wedge (∀y, β \wedge (∃ u, ∀ v, ¬y))]$

(D) choice

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dont solve full its just waste of time 

follow my rule and save your time

 

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