## Negation and the truth of propositions

First form the negation of each of the following propositions. Then decide whether the original proposition or its negation is true.

• #### Variant 1

$$\forall x,y\in \mathbb R: x^2+y^2>0$$

• #### Variant 2

$$\forall x\in \mathbb R\ \exists n\in \mathbb N: x< n$$

• #### Variant 3

$$\forall x\in \mathbb R\ \exists n\in \mathbb N: (x\ge n) \land (x<n+1)$$

• #### Variant 4

$$\forall \varepsilon> 0\ \exists \delta>0\ \forall x\in \mathbb R: |x-2|<\delta \ \Rightarrow\ |x-3| < \varepsilon$$

• #### Variant 5

$$\forall \varepsilon> 0\ \exists \delta\ge 0\ \forall x\in \mathbb R: |x-2|<\delta \ \Rightarrow\ |x-3| < \varepsilon$$
(this differs from the previous variant in the range of $$\delta$$)

• #### Variant 6

$$\forall x\in \mathbb N\ \exists y\in \mathbb N\ \forall z\in \mathbb N: z>x \Rightarrow y<z$$

• #### Variant 7

$$\exists y\in \mathbb N\ \forall x\in \mathbb N\ \forall z\in \mathbb N: z>x \Rightarrow y<z$$

• #### Variant 8

$$\exists y\in \mathbb N\ \forall x\in \mathbb N\ \forall z\in \mathbb R: z>x \Rightarrow y<z$$

• #### Variant 9

$$\exists y\in \mathbb N\ \forall x\in \mathbb R\ \forall z\in \mathbb N: z>x \Rightarrow y<z$$