Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 1

Exercises

Exercise

Which of the following statements are scientific facts? To which science do they belong? On what does its validity or invalidity rest?

1. Smoking is dangerous for the health.
2. Dinosaurs became extinct some 65 million years ago.
3. Dinosaurs did not die out at all.
4. Such a thing as a dinosaur never existed.
5. The gospels were written by eyewitnesses.
6. The ${}abc$-conjecture is now a theorem.
7. The ${}abc$-conjecture is still a conjecture.
8. The theory of relativity is confirmed.
9. It is not possible to make gold out of other materials.
10. The world will soon perish.

Exercise

In the lecture, we indicated a quite positive view on science. There are also many quite opposite opinions, as expressed in the following statements. What is your opinion?

1. Science is a tool for authority and dominance.
2. Science is just a loony occupation of a small elite.
3. Science is a modern fairy tale, a construct of language, a discourse, a narrative, which could also be deconstructed.
4. Science serves alone to the maintenance of the patriarchy.
5. Science is against god.
6. Science is just an arbitrary collection of statements, the opposite is also true.
7. The so-called science yields only a very superficial idea of the world. True knowledge emerges in uniting with the world.

Exercise

Why is mathematics difficult, though anything in it is logical?

Exercise

Paraphrase the following statements as If-then-statements.

1. What little John does not learn, John will never learn.
2. What the farmer does not know, he will not eat.
3. As soon as the sun is shining, Lucy steps outside.
4. Beginning with ${}32$ marks, you get a first degree.
5. With this attitude, you should not become a teacher.
6. What does not kill us makes us stronger.
7. Start at an early age in order to become a master.
8. You cannot say ${}A$ without saying ${}B$.
9. Who does not arrive for dinner in time, has to look what is left over.
10. Let anyone among you who is without sin be the first to throw a stone at her.

Exercise

Write the contrapositions to the statements formulated in Exercise 1.4 . Avoid double negations.

Exercise

Formalize the statement "If the wind of change is blowing, some build walls and some build windmills“ with propositional variables and with logical connectives.

Exercise

Suppose that the following implications are known.

1. If Mustafa Müller makes funny grimaces, then Heinz Ngolo has to hold his stomach.
2. If he eats too many gummy bears, then Heinz Ngolo has to hold his stomach.
3. If he receives a ball against his stomach, then Heinz Ngolo has to hold his stomach.

At this moment, Heinz Ngolo does not hold his stomach. What can you deduce from this information?

Exercise

The following facts hold: If we are not in vacation and unless it is weekend and if he is not ill, then Heinz Ngolo has to go to school. Today, Heinz Ngolo need not go to school. What can you deduce from this?

Exercise

The following facts are known.

1. The earthworms are glad if and only if it rains or snows.
2. The children are glad if and only if the sun is shining or it snows.

What conclusion can we draw in each of the following cases.

a) The children and the earthworms are glad.

b) The children are glad and the earthworms are not glad.

c) The children are not glad and the earthworms are glad.

Exercise

The following facts are known.

1. The early bird catches the worm.
2. Doro is not caught by Lilly.
3. Lilly is a bird or a hedgehog.
4. For a hedgehog, 5 o'clock in the morning is late.
5. Doro is a worm.
6. For a bird, 5 o'clock in the morning is early.
7. Lilly sleeps until 5 o'clock in the morning and after that she is on its way.

Answer the following questions.

1. Is Lilly a bird or a hedgehog?
2. Is she an early or a late animal?
3. Catches the late hedgehog the worm?

Exercise

In the football cup, Bayern München plays versus TSV Wildberg. The coach of TSV Wildberg, Mr. Tor Acker, says "We have nothing to lose in this match“. The teacher of logic in Wildberg, Ms. Loki Schummele, says "If Wildberg has nothing to lose in this match, then also München has nothing to win in this match“. The coach of Bayern München, Mr Roland Rollrasen, says "We have in this game something to win“.

1. Is the statement of Ms. Schummele logically correct?
2. Assume that the statement of the coach of Bayern is true. What conclusion can we draw from the statement of Mr. Acker?

Exercise

Prove, using truth tables, that the following statements are tautologies.

1. ${}\alpha \wedge \beta \longleftrightarrow \beta \wedge \alpha$.
2. ${}\alpha \vee \beta \longleftrightarrow \beta \vee \alpha$.

Exercise

Prove, with the help of truth tables, that the following statements are tautologies.

1. ${}{\left(\alpha \wedge \beta \right)}\wedge \gamma \longleftrightarrow \alpha \wedge {\left(\beta \wedge \gamma \right)}$.
2. ${}{\left(\alpha \vee \beta \right)}\vee \gamma \longleftrightarrow \alpha \vee {\left(\beta \vee \gamma \right)}$.

Exercise

Prove, with the help of truth tables, that the following statements are tautologies.

1. ${}(\alpha \wedge \alpha )\leftrightarrow \alpha$.
2. ${}\alpha \wedge \beta \rightarrow \alpha$.
3. ${}\alpha \rightarrow {\left(\beta \rightarrow \alpha \right)}$.
4. ${}{\left(\alpha \rightarrow {\left(\beta \rightarrow \gamma \right)}\right)}\rightarrow {\left({\left(\alpha \rightarrow \beta \right)}\rightarrow {\left(\alpha \rightarrow \gamma \right)}\right)}$.
5. ${}{\left(\alpha \rightarrow \beta \right)}\leftrightarrow (\neg \alpha \vee \beta )$.

Exercise

Prove, with help of truth tables, the De Morgan's laws, namely that the statements

$\neg {\left(\beta \vee \gamma \right)}\leftrightarrow {\left(\neg \beta \wedge \neg \gamma \right)}$

and

$\neg {\left(\beta \wedge \gamma \right)}\leftrightarrow {\left(\neg \beta \vee \neg \gamma \right)}$

are tautologies.

Exercise

Prove, with help of truth tables, the (generalized) De Morgan's laws, namely that the statements

${\left(\alpha \wedge \neg {\left(\beta \vee \gamma \right)}\right)}\leftrightarrow {\left({\left(\alpha \wedge \neg \beta \right)}\wedge {\left(\alpha \wedge \neg \gamma \right)}\right)}$

and

${\left(\alpha \wedge \neg {\left(\beta \wedge \gamma \right)}\right)}\leftrightarrow {\left((\alpha \wedge \neg \beta )\vee {\left(\alpha \wedge \neg \gamma \right)}\right)}$

are tautologies.

Exercise

Show that the propositional expression

${\left(r\rightarrow {\left(p\wedge \neg q\right)}\right)}\rightarrow {\left(\neg p\rightarrow {\left(\neg r\vee q\right)}\right)}$

is always valid.

Exercise

Find a quite simple expression from propositional logic, which describes the following truth function, given by a truth table.

${}p$ ${}q$ ${}?$ t t f t f f f t t f f f

Exercise

Find a quite simple expression from propositional logic, which describes the following truth function, given by a truth table.

${}p$ ${}q$ ${}?$ t t t t f t f t f f f t

Exercise

Stuck in a cave, there are four people at the end of the corridor. They have just one flashlight, and the corridor leading outside can only be tackled with the flashlight. At most two people can walk together through the corridor. The people have different levels of skills to walk through the corridor, the first person needs an hour, the second person needs two hours, the third person needs four hours and the fourth person needs five hours to pass the corridor. If two of them walk together, always the slower one determines the velocity.

1. Suppose that the battery of the flashlight works for exactly ${}13$ hours. Can the four leave the cave?
2. Suppose that the battery of the flashlight works for exactly ${}12$ hours. Can the four leave the cave?

A natural number ${}n$ is called even, if it can be written in the form ${}n=2k$

with some natural number ${}k$.

A natural number is called odd, if it is not even. Try to prove the following familiar statements from this definition. What do you have to know about the decimal expansion of a number?

Exercise

Show that a natural number ${}n$ is even if and only if its last digit in the decimal system equals ${}0,2,4,6$ or ${}8$.

Exercise

Show that a natural number ${}n$ is odd if and only if its last digit in the decimal system equals ${}1,3,5,7$ or ${}9$.

Exercise

Show that a natural number ${}n$ is odd if and only if it is of the form ${}n=2k+1$ with some natural number ${}k$.

Exercise

Let ${}n$ denote a natural number. Show, by considering cases, that ${}n^{2}-n$ is always even.

Hand-in-exercises

Exercise (4 marks)

The following statements are known.

1. In summer vacation, we are going to Italy.
2. In winter vacation, we are going to Austria.
3. If we are in Austria, then we visit are grandma.
4. If we go to Italy, then we drive via Switzerland or Austria.

Answer the following questions.

a) We drive to Italy, but not via Switzerland. Do we visit grandma?

b) We are in Summer vacation and we do not drive through Switzerland. Do we visit grandma?

c) Is it possible to deduce the statement "if we do not visit grandma, then we are not in winter vacation“ from the conditions?

d) Is it possible to deduce the statement "in the summer vacation and in the winter vacation we visit grandma“ from the conditions?

Exercise (2 marks)

Determine the truth value of the statement

$(((\neg (\neg (p)))\rightarrow (\neg (q)))\vee (\neg (r)))\leftrightarrow ((\neg (r))\wedge (q)),$

if ${}p$ and ${}r$ are false and ${}q$ is true.

Exercise (2 marks)

Prove, with the help of truth tables, that the following statements are tautologies.

1. ${}{\left(\alpha \wedge {\left(\beta \vee \gamma \right)}\right)}\longleftrightarrow {\left(\alpha \ \wedge \beta \right)}\vee {\left(\alpha \wedge \gamma \right)}$.
2. ${}{\left(\alpha \vee {\left(\beta \wedge \gamma \right)}\right)}\longleftrightarrow {\left(\alpha \vee \beta \right)}\wedge {\left(\alpha \vee \gamma \right)}$.

Exercise (2 marks)

This is how our artist thinks about a Zombie in a good mood.

We consider the following quotation of Sven Walter from the article "Zombies, Dualismus und Physikalismus“ (Zeitschrift for philosophische Forschung, Bd. 65, H. 2 (2011), pp. 241-254, https://www.jstor.org/stable/pdf/41346224.pdf).

" (${}P_{1}$) Zombies are imaginable.

(${}P_{2}$) If Zombies are imaginable, then Zombies are possible.

(${}P_{3}$) If Zombies are possible, then the physicalism is wrong.

Hence: the physicalism is wrong.“

Formalize the propositional rules used here, and show, with the help of truth tables, that they are tautologies.

Exercise (4 marks)

The gang of robbers called "Robin Hood“ consists of five people. They hide their stolen goods in a treasure chest. They want to secure the chest with several locks, and they want to distribute the (multiple) keys to the members. The distribution should be such that any two members alone are not able to open the chest, but any three members are able to open the chest. How many locks do they need, and how do they have to distribute the keys?

Exercise (3 (1+1+1) marks)

1. Formulate computing rules for the addition and the multiplication of even and odd natural numbers.
2. Prove these rules with the last-digit-description (see Exercise 1.21 and Exercise 1.22 ).
3. Prove these rules with the description with equations (definition and Exercise 1.23 ).

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