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# 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.

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

## Exercise ​

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

StatementNegation
or Not and not
and Not or not
If , then and not
For all , There exist such that not
There exists such that For every , not

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

## Exercise ​

Find the negation of the following statements

1. Some prime numbers are odd.
2. Nobody is lazy.
3. Some horses are black.
4. For every , .
5. There exists such that .
6. For each there exists such that .
7. If then implies that .

## Exercise ​

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

1. For every positive number , and every positive number , we have .
2. There exists a positive number such that for every positive number , we have .
3. There exists a positive number , and there exists a positive number , such that .
4. For every positive number , there exists a positive number such that .
5. 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?