# Additional exercises 1

## Exercise

Identify the hypothesis and conclusion in the following statements. For the 4th and 5th statements, rewrite them in a way that clarifies which part is hypothesis and which is conclusion.

- If
is a positive real number that satisfies , then . - If
is a real number, then either , . - If there exists a real number
such that , then any real number is either an integer or a multiple of . - The inverse of a irrational number is also irrational.
- For each
there exists such that .

## Exercise

The negation of various statements is described in the following table.

Statement | Negation |
---|---|

Not | |

Not | |

If | |

For all | There exist |

There exists | For every |

Think of day-to-day examples which illustrate the logic described in this table.

## Exercise

Find the negation of the following statements

- Some prime numbers are odd.
- Nobody is lazy.
- Some horses are black.
- For every
, . - There exists
such that . - For each
there exists such that . - If
then implies that .

## Exercise

What does each of the following statements mean, and which of them are true?

- For every positive number
, and every positive number , we have . - There exists a positive number
such that for every positive number , we have . - There exists a positive number
, and there exists a positive number , such that . - For every positive number
, there exists a positive number such that . - There exists a positive number
such that for every positive number , we have .

## Exercise

Consider the following statement.

Let

be differentiable on and suppose that takes a local minimum or maximum at the point . Then .

Discuss the possibility to improve the statement in the sense of making it stronger. Would this be stronger or weaker conclusion? Would this be a stronger or a weaker hypothesis?