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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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[∃x,α→(∀y,β→(∃u,∀v,y))]

here α and β are independent of x, y or here it is not given that α and β are function of x and y then how to find negation in such case
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1 Answer

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50 votes
Best answer

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

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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4 Comments

@Mk Utkarsh

I think "," or comma is used to separate 2 premises
 

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So, we've to treat "," as "()" always in any 1st order logical expression???
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Why is option B not correct @Arjun Sir? I am getting both B and D. 

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Answer:

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