# Math Help - Negation of a sentence

1. ## Negation of a sentence

I need to write the negation for each of the following. I've written out what I think it should be, but was hoping someone could check. I want to be sure I understand this.

$\forall x \in Z, x < 0$

Negation: $\exists x \in Z, x \not< 0$

$\exists x \in Z, x = x + 1$

Negation: $\forall x \in Z, x \neq x + 1$

$\exists x \in N, x < 10$

Negation: $\forall x \in N, x \not< 10$

$\forall x \in N, x + x = 2x$

Negation: $\exists x \in N, x + x \neq 2x$

$\exists x \in Z, \forall y \in Z, x > y$

Negation: $\forall x \in Z, \exists y \in Z, x \not> y$

$\forall x \in Z, \exists y \in Z, x = y$

Negation: $\exists x \in Z, \forall y \in Z, x \neq y$

$\forall x \in Z, \exists y \in Z, x + y = 0$

Negation: $\exists x \in Z, \forall y \in Z, x + y \neq 0$

Would it be more proper to use the $\neg$ symbol, or are these okay?

2. those all look good to me, and I don't use that symbol but then again I am not really dealing with logic tables too much

3. Thanks