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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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