1. ## Quantifer Negation Help

I have been reading on how to write the negation of quantifiers for a couple hours but am still confused as to how to give the answer:

Give the Negation of the following:
1. $\displaystyle \forall z \in \mathbb{C}, \exists w \in \mathbb{C} : w^2 = z$

2. $\displaystyle \forall x \in \mathbb{R}, (x<-2 \rightarrow x^2 >4)$

1. $\displaystyle \exists z \in \mathbb{C}, \forall w \in \mathbb{C} : \neg (w^2 = z)$

2. $\displaystyle \exists x \in \mathbb{R}, \neg (x<-2 \rightarrow x^2 >4)$

2. Yeah, you are correct.

3. Originally Posted by Stylis10
I have been reading on how to write the negation of quantifiers for a couple hours but am still confused as to how to give the answer:

Give the Negation of the following:
1. $\displaystyle \forall z \in \mathbb{C}, \exists w \in \mathbb{C} : w^2 = z$

2. $\displaystyle \forall x \in \mathbb{R}, (x<-2 \rightarrow x^2 >4)$

1. $\displaystyle \exists z \in \mathbb{C}, \forall w \in \mathbb{C} : \neg (w^2 = z)$
2. $\displaystyle \exists x \in \mathbb{R}, \neg (x<-2 \rightarrow x^2 >4)$
1. $\displaystyle \exists z \in \mathbb{C}, \forall w \in \mathbb{C} : w^2 \ne z)$
2. $\displaystyle \exists x \in \mathbb{R}, (x<-2 ~\&~ x^2 \le4)$