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

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

Difficulty level: Moderate task
Proving or derivation task
Cs translation
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