# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 3

*Exercises*

Determine for the sets

the following sets.

- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .

Let denote the set of capital letters in the Latin alphabet, the set of capital letters in the Greek alphabet and the set of capital letters in the Russian alphabet. Determine the following sets.

- .
- .
- .
- .
- .

Let and denote sets. Prove the identity

Let and denote sets. Prove the following identities.

Prove the following (settheoretical versions of) syllogisms of Aristotle. Let denote sets.

- Modus Barbara: and imply .
- Modus Celarent: and imply .
- Modus Darii: and imply .
- Modus Ferio: and imply .
- Modus Baroco: and imply .

Does the "subtraction rule“ hold for the union of sets, i.e., can we infer from that holds?

Let denote sets. Show that the following statements are equivalent.

- .
- .

Sketch the product set as a subset of .

Describe for any choice of two of the following geometric sets its product set (including the case where a set is taken twice).

- A line segment .
- A circle (circumference) .
- A disk .
- A parabola .

Which of these product sets can be realized within space, which can't?

Sketch the following subsets in .

- ,
- ,
- ,
- .

- Sketch the set and the set .
- Determine the intersection geometrically and arithmetically.

We recommend illustrating the formulated set identities of the following exercises.

Let and denote sets and let and be subsets. Show the identity

Let and denote sets and let and be subsets. Show the identity

Let be a set of people and the set of the first names and the set of the surnames of these people. Define natural mappings from to , to and to and discuss them using the relevant notions for mappings.

Determine for the following diagrams which empirical mappings they describe. What is in each case the source, the target, which units are used? Does each image represent one or more mappings? Do they really represent mappings? What information is conveyed beyond the mapping? Is there some kind of mathematical modelling for these empirical mappings?

Give examples of mappings

such that is injective, but not surjective, and is surjective, but not injective.

Show that there exists a bijection between and .

Establish, for each , whether the function

is injective and/or surjective.

Which graphs of a function do you know from school?

How can we recognize by looking at the graph of a mapping

whether is injective or surjective?

Which bijective functions (or between subsets of ) do you know from school? What is the name of the inverse function?

Let and denote sets. Show that the mapping

is a bijective mapping between the product sets and .

Let and be sets and let

be a function. Let

be another function such that and . Show that is the inverse of .

We consider the sets

and the mappings and which are given by the value tables

and

respectively.

- Determine a value table for .
- Are the mappings , , injective?
- Are the mappings , , surjective?

Determine the composite functions and for the functions , defined by

- Can a constant mapping be bijective?
- Is the composition of a constant mapping with an arbitrary mapping (the constant map first) constant?
- Is the composition of an arbitrary mapping with a constant mapping (the constant map last) constant?

Let and be sets and let

and

be functions. Show that

Let and denote sets and let

and

be mappings with the composition

Show the following properties.

- If and are injective, then also is injective.
- If and are surjective, then also is surjective.
- If and are bijective, then also is bijective.

Let be sets and let

be functions with their composition

Show that if is injective, then also is injective.

*Hand-in-exercises*

### Exercise (4 marks)

Let and be sets. Show that the following facts are equivalent.

- ,
- ,
- ,
- There exists a set such that ,
- There exists a set such that .

### Exercise (2 marks)

Sketch the following subsets in .

- ,
- ,
- ,
- .

### Exercise (3 marks)

Let be sets and let

be functions with their composite

Show that if is surjective, then also is surjective.

### Exercise (4 marks)

We consider a computer with only two memory units, each can represent a natural number. At the start of every program (a sequence of commands), the initial entries are . The computer can empty a memory unit, it can increase a memory unit by , it can jump to another command (unconditional Goto) and it can compare the two entries of the two memory units. Moreover, it can jump to another command in case the comparing condition is fulfilled (conditional Goto). There is a printing command which prints the entries at the moment. Write a computer-program such that every pair is printed exactly once.

### Exercise (4 marks)

Determine the compositions and for the mappings given by

### Exercise (3 marks)

Consider the set and the function

defined by the following table

Compute , that is the -rd composition (or iteration) of with itself.

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