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I faced a question in exam where we had to tick the correct negation of the following statement

$"\exists y\in \Bbb{Z}, \forall x\in \Bbb{R}, \text{such that}\ y^2<x"$.

I checked the option $"\forall\ y\in \Bbb{Z}, \exists\ x\in \Bbb{R}, \text{such that}\ y^2\ge x"$.

Does it seem correct to you? What is a general rule while negating statements involving $\exists$ and $\forall$ symbols?

blabla
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    Looks correct to me. I think you appliead all the correct rules. Switching $\forall$ with $\exists$ and negating the inner statement. – M. Winter May 17 '17 at 12:33

1 Answers1

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Yes, that is correct. Here are the general rules involving quantifiers and negation:

Quantifier Negation

For any formula $\varphi$:

$\neg \forall x \varphi \Leftrightarrow \exists x \neg \varphi$

$\neg \exists x \varphi \Leftrightarrow \forall x \neg \varphi$

If we apply this to your statement, we get:

$\neg \exists y\in \Bbb{Z} \ \forall x\in \Bbb{R} \ y^2<x \Leftrightarrow$

$\forall y \in \Bbb{Z}\ \neg \forall x\in \Bbb{R}\ y^2<x \Leftrightarrow$

$\forall y \in \Bbb{Z}\ \exists x\in \Bbb{R}\ \neg y^2<x \Leftrightarrow$

$\forall y \in \Bbb{Z}\ \exists x\in \Bbb{R}\ y^2 \ge x $

Bram28
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