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I understand that for negation $\forall$ converts into $\exists$

like: $\forall x \ , \forall y$, $ \ y^2 + x^2 \ge0$

and the negation will be: $\exists x \ $and $ \exists y$, $ \ y^2 + x^2 < 0$

I understand that it translates to: there exists an element of $x$ and of $y$ that $y^2 + x^2 < 0$ which make sense, but:

My Question

For this particular case, could this be another form of negation: $\forall x \ , \forall y$, $ \ y^2 + x^2 <0$

Wouldn't this be a stronger statement than the "correct" one, or is it too strong/absolute that it might miss a particular case that is not necessary $\forall x \ , \forall y$?

I would really appreciate any insight you can offer

Reuben
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    That's much stronger than the negation. The negation of "it rained every day last month" is "there was at least one day without rain last month" not "it didn't rain even once last month". – lulu Apr 20 '22 at 18:01
  • @lulu ok I think I understand, negation must be "simplest" form that contradicts the statement – Reuben Apr 20 '22 at 18:16
  • It's effectively a stronger statement since you take $x,y\in \mathbb{R}\neq\emptyset$. Also, remark that your "other negation" is the negation of $\exists x,\exists y$ such that $x^2+y^2\geq 0$ which is weaker than $\forall x,\forall y$, $x^2+y^2\geq 0$. – SacAndSac Apr 20 '22 at 18:22
  • @Reuben It's not accurate to say that a negation is the "simplest" statement form that contradicts the original. The negation of "every couple lies on/outside the unit circle" is "some couple lies within the unit circle" instead of "every couple lies within the unit circle", which simply does not logically flip the original's statement's T/F value. For quantified statements, negation is not quite analogous to taking complement. Maybe relevant: https://math.stackexchange.com/a/4424427/21813 – ryang Apr 20 '22 at 18:49
  • As others have remarked, it's not a question of simplicity. To declare that the negation of a statement is true is the same as declaring that the statement is false. Knowing that "it rained every day last month" is false only tells you that there was some day without rain last month. Not that there was no rain at all last month. – lulu Apr 20 '22 at 19:12
  • It just occurred to me that "within" might be interpreted as including the boundary; as such, please replace "within the unit circle" with "inside the unit circle" in my above example. – ryang Apr 21 '22 at 13:39

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