# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 4

*Exercises*

Show that the composition on a line which assigns to two points their midpoint, is commutative, but not associative. Does there exist a neutral element?

Let be the set of all female compound first names (names of the form , where are simple first names and every combination is allowed). We consider the composition

which sends a pair of double first names to the double name .

- What is the value of under this composition?
- Is this composition commutative?
- Is this composition associative?
- Does this composition have a neutral element?
- Is this composition surjective?
- Is this composition injective?

Let be an element in a field . Show that the element fulfilling is uniquely determined.

Let be a field and let be elements from . Show that the following rules for sign hold.

Describe and prove rules for the addition and the multiplication of even and odd integer numbers. Define on the set with two elements

an "addition“ and a "multiplication“ which "represents“ these rules.

Let be a field. Show that for every natural number there exists a field element such that is the null element in and is the unit element in and such that

holds. Show that this assignment has the properties

Extend this assignment to all integer numbers and show that the stated structural properties hold again.

Let be a field with . Show that for the relation

holds.

Let be elements in a field and suppose that and are not zero. Prove the following fraction rules.

Does there exist an analogue of formula (8), which arises when one replaces addition by multiplication (and subtraction by division), that is

Show that the popular formula

does not hold.

Let be a field and let be elements in . Prove the following laws for powers for natural exponents .

Let be a field and let be elements in . Prove the following laws for powers for integer exponents . The corresponding laws for exponents from can be used and also the facts that the inverse of the inverse is the element itself and that the inverse of equals .

Compute the matrix product

a) Give an example of rational numbers such that

b) Give an example of rational numbers such that

c) Give an example of irrational numbers and a rational number such that

Show that the binomial coefficients satisfy the following recursive relation

Show that the binomial coefficients are natural numbers.

Prove the formula

Show by induction that for the estimate

holds.

Franziska wants to break up with her boyfriend Heinz. She thinks about the following explanations.

- "You behaved already on the very first day completely weird. Since then, from each day to the next day, you got worse. Hence, you will continue to behave completely weird“.
- "If I would stay together with you, I will end up as a sad bored person disappointed with life, and this I definitely do not want“.
- "Well, if you do not love me, then I do not want you anyway. But if you love me, then I come to the conclusion that you are not able to behave in a way which fits your feelings. But then you are not mature, and then I do not want you neither“.

Which mathematical proof principles can be seen in these explanations?

- Solve the following mini-sudoku
- Show that the mini-sudoku in (1) has only one solution.
- Which mathematical proof principles can be found as typical argumentation patterns when solving a sudoku?

Look at the following statement: "The principle "Proof by contradiction“ is obviously nonsense. If we can assume anything, then we can always arrive at a contradiction and then we can prove anything“. Discuss this statement.

*Hand-in-exercises*

### Exercise (2 marks)

Show that taking powers of positive natural numbers, i.e., the assignment

is neither commutative nor associative. Does this composition have a neutral element?

### Exercise (2 marks)

Let be an element in a field different from . Show how to compute with four multiplications.

### Exercise (5 marks)

Prove the general distributive property for a field.

### Exercise (3 marks)

Show that the "rule“

is for (and ) never true. Give an example with where this rule holds.

### Exercise (2 (1+1) marks)

Let and .

- Show that is an irrational number if and only if is irrational.
- Suppose now also that . Show that is irrational if and only if is irrational.

### Exercise (3 marks)

Prove the following formula

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