## Negation and the truth of propositions

### Task number: 2783

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

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