# Discrete Mathematics DSBA2020/2021

## Содержание

## Exam

We shall have the final exam on April, 2. An attendee of the exam can bring some printed or handwritten materials on paper that may help in solving the given problems. All other sources of information apart from the attendee's brain are **strictly prohibited**.

Exam starts at 16:00 MSK and takes place in rooms R401 and R405. For distribution of students into rooms please check the exam sheet of the table.

The list of students approved for online exam.

Year 2020 variant of the exam with the solution.

## Colloquium

Please follow this link for the Colloquium Rules and Question List.

Get the colloquium cards for Group #203 by this link (December, 11)

The Colloquium schedule is as follows:

- Group #201 - 16:20 MSK, December, 10
- Group #202 - 18:10 MSK, December, 10
- Group #203 - 16:20 MSK, December, 11
- Group #204 - 16:20 MSK, December, 9

The links to the conferences will appear there:

- Ivanov Anton.
- Danilov Boris.
- Karimov Rustam.
- Ivanova Darya.
- Trofimova Anastasia.
- Fishman Maxim.
- Evgeny Dashkov.

The list may update in the future

## Test

We shall have a written test on November, 24. The test is scheduled for 6:10 pm MSK. You are supposed to send you work via a Google-form.

## Current Results

You can check your progress up to now via the DM1 Register Online.

## Homework Deadlines

**For Group 201:**

- HW 1, Tasks 1--5: September 18
- HW 1, Tasks 6--9: September 25
- HW 4, Tasks 1--13: January, 31
- HW 5, Tasks 1--16: March, 1
- HW 6, Tasks 1--10: March, 20
- HW 7, Tasks 1--14: March, 28

**For Group 202:**

- HW 1, Tasks 1--3: September 15
- HW 1, Tasks 4--5: September 22
- HW 1, Tasks 7--9: September 28
- HW 2, Tasks 1--3: October 13
- HW 2, Tasks 4--9: October 27
- HW 2, Tasks 10--12: November, 2
- HW 2, Tasks 13--22: November, 19
- HW 3, Tasks 1--5: December, 1
- HW 3, Tasks 6--12: December, 15
- HW 4, Tasks 1--13: January, 31
- HW 5, All tasks: March, 11
- HW 6, All tasks: March, 18
- HW 7, All tasks: March, 28

**For Group 203:**

- HW 1, Tasks 1--3: September 15
- HW 1, Tasks 4--5: September 22
- HW 1, Tasks 7--9: September 28
- HW 2, Tasks 1--3: October 13
- HW 2, Tasks 4--9: October 27
- HW 2, Tasks 10--11: November, 2
- HW 2, Tasks 12--22: November, 19
- HW 3, Tasks 1--5: December, 1
- HW 3, Tasks 6--12: December, 15
- HW 4, Tasks 1--13: January, 31
- HW 5, All tasks: March, 11
- HW 6, All tasks: March, 18
- HW 7, All tasks: March, 28

**For Group 204:**

- 1. HW 1, Tasks 1--3: September 16
- 2. HW 1, Tasks 4: September 23
- 3. HW 1, Tasks 5-6: September 30
- 4. HW 1, Tasks 7-9: October 7
- 5. HW 2, Tasks 1-3: October 14
- 6. HW 2, Tasks 4-10: October 23
- 7. HW 2, Tasks 11-15: November 7
- 8. HW 2, Tasks 16, 17, 21, 22: November 15
- 9. HW 2, Tasks 18-20: November 19
- 10. HW 3, Tasks 1-5: December 5
- 11. HW 3, Tasks 6-12: December 19
- 12. HW 4, Tasks 1-9: January 30
- 13. HW 4, Tasks 10-13: February 6
- 14. HW 5, Tasks 1, 4, 6, 10 : February 16
- 15. HW 5, Tasks 2,5,8,9,13: February 20
- 16. HW 5, Tasks 7,12: February 27
- 17. HW 5, Tasks 3, 11, 14-16: March 6
- 18. HW 6: March 20
- 19. HW 7: March 28

## Course Materials

### Lecture Notes

You can find some useful materials (including the Lecture Notes) here.

### Problem sets

Class Problems | Homework Assignments |
---|---|

cw1 | hw1 |

cw2 | hw2 |

cw3 | --- |

cw4 | --- |

cw5 | --- |

cw6 | hw3 |

cw7 | --- |

cw8 | hw4 |

cw9 | hw5 |

cw10 | hw6 |

cw11 | hw7 |

cw12 | hw8 |

### Videos

#### Lectures

Lecture 8: The Extended Euclidean Algorithm; solving linear Diophantine equations in two variables.

Lecture 9: Solving linear congruence; the Chinese Remainder Theorem; practical solving of simultaneous congruences.

Lecture 10: Sets: an axiomatic approach; elementhood relation; set inclusion and equality; the Axioms of Foundation and Equality; the "Substitution Principle"; set construction principles; grouping a few sets together; singletons; specifying a subset; the empty set; powersets; the union of two sets (and a more general case); the intersection and difference of two sets.

Lecture 11: Expressing inclusion in terms of equality; set algebra identities (with application examples); (Kuratowski's) ordered pair; pair equality criterion; Cartesian product; ordered tuples; Cartesian power.

Lecture 12: Binary relation between two sets; the domain, range, and field of a relations; relation diagram; relation on a set; the identity relation; the converse relation; the converse of the converse and of the union of relations; the composition of two relations; composition associativity; the converse of the composition; the composition with the identity; set image and pre-image under a relation.

Lecture 13: Functional, injective, total, and surjective relations; these classes under conversion and composition; functions and their values; the composition of two functions; function equality criterion; injections, surjections, and bijections.

Lecture 14: bijectivity criterion; Galileo's 'paradox'; set equivalence and its first properties; \N^2 is equivalent to \N; Cantor's Theorem.

Lecture 15: Identities for set products and exponentials; indicator functions and subsets.

Lecture 16: Ordered tuples and finite domain functions; embeddings; Cantor--Schroeder--Bernstein Theorem with applications; Continuum Hypothesis.

Lecture 17: Finite and infinite sets; the Pigeonhole Principle and related results; the cardinality (size, number of elements) of a finite set.

Lecture 18: Subsets of a countable set; a countable set is minimal infinite; the Rules of Sum and Product; counting tuples, functions, and subsets for finite sets; a countable set is the least infinite set.

Please find the **later recordings** here.

#### Seminars by Evgeny Dashkov

Seminar 11. Sets.

Seminar 12. Set algebra identities. Cartesian product.

Seminar 13. Binary relations. Converse relation, composition, set algebra operation. Set image under relation.

Seminar 14. Images and pre-images. Funtional, injective, total, and surjective relations.

Seminar 15. Functions, injections, surjections, bijections.

Seminar 16. A continuation of the above; indicator functions.

Seminar 17. Cantor--Schroeder--Bernstein Theorem.

Seminar 18. Cantor--Schroeder--Bernstein Theorem. Countable sets. The Pigeonhole Principle.

Please find the **later recordings** here.

#### Miscellaneous

Although we had no regular recording of lectures or seminars until Module 2, a few videos on selected topics are available.

#### Video 1

An equivalence proof for the Strong Induction, Mathematical Induction and Least Number Principles.

#### Video 2

The definition of the greatest common divisor (GCD); the uniqueness and existence theorem.

#### Video 3

A compensatory seminar for Group 201. The main topics are GCD and LCM.

#### Video 4

A supplementary material partly covered at the extra class with groups 202 and 203 on October, 30. The main topics are common divisors and common multiples, solutions to exercises 4, 5, 7-9 from the classwork sheet #4.

Watch it! and get the whiteboard!

## Other Resources

- It is HIGHLY recommended to join our Telegram chat.

- We have a dedicated server to hold an online meeting if we need one.

- We encourage everyone to join Google Classroom for this course: there is a global classroom for everyone and group-specific classrooms. Connect using class codes below:

For all groups | Group 201 | Group 202 | Group 203 | Group 204 |
---|---|---|---|---|

civetu3 | 7pmgfzr | nf3kg65 | 36dtkir | 76a5te5 |

## Professors

### The Lecturer

My name is Evgeny Dashkov. Feel free to contact me via email: edashkov@gmail.com, Telegram: @edashkov, or VK.

### Technical Support

My name is Boris Danilov. Please, address me all issues and questions related to distant learning technologies that we use in our course. My contacts: brdanilov@gmail.com, Telegram @brdann .

### Seminar Instructors

- Group 201: Evgeny Dashkov
- Group 202: Boris Danilov
- Group 203: Boris Danilov
- Group 204: Anastasia Trofimova

### Teaching Assistants

- General TA: Anton Ivanov. Contact via Telegram https://t.me/Olenek53
- TA of group 201: Darya Ivanova. Contact via Telegram https://t.me/ivanovskayaaaaa
- TA of group 202: Rustam Karimov. Contact via Telegram https://t.me/dergalaktischekaiser
- TA of group 203: Anton Ivanov. Contact via Telegram https://t.me/Olenek53
- TA of group 204: Maxim Fishman. Contact via Telegram https://t.me/co_ban

## Recommended Reading

*Please notice that The Book for our Course does not exist. The latter is based on many sources.*

- Anderson J. A., Discrete Mathematics With Combinatorics. Prentice Hall, 2003.
- Biggs N. L., Discrete mathematics. 2nd ed., New York; Oxford: Oxford University Press, 2004.
- Gavrilov G. P., Sapozhenko A. A. Problems and Exercises in Discrete Mathematics. Kluwer Texts in the Mathematical Sciences 14. Springer, 1996.
- Lehman E., Thomson Leighton F., Meyer A. R. Mathematics for Computer Science, 2017.
- Lovasz L., Vesztergombi K. Discrete Mathematics. Lecture Notes; Yale University, 1999.
- Melnikov O., Sarvanov V., Tyshkevich R., Yemelichev V., Zverovich I. Exercises in Graph Theory. Kluwer Texts in the Mathematical Sciences 19. Springer, 1998.
- Rosen K. H. Discrete Mathematics and Its Applications. McGraw-Hill, 1999.
- Stein C., Drysdale R. L., Bogart K. Discrete mathematics for computer scientists. Addison-Wesley, 2010.
- Vinogradov I. M. Elements of number theory. Dover, 1954.

#### In Russian

*If you understand Russian (by any chance), you will probably benefit from reading the following books.*

- Виноградов И. М. Основы теории чисел. 9-е изд., М.: Наука, 1981.
- Вялый М., Подольский В., Рубцов А., Шварц Д., Шень А. Лекции по дискретной математике.
- Гаврилов Г. П., Сапоженко А. А. Задачи и упражнения по дискретной математике. 3-е изд., М.: ФИЗМАТЛИТ, 2004.
- Дашков Е. В. Введение в математическую логику. Множества и отношения. М.: МФТИ, 2019.
- Зубков А. М., Севастьянов Б. А., Чистяков В. П. Сборник задач по теории вероятностей. 2-е изд., М.: Наука, 1989.
- Мельников О. И. Теория графов в занимательных задачах. 5-е изд., М.: Книжный дом "ЛИБРОКОМ", 2013.
- Шень А., Математическая индукция. 5-е изд, М.: МЦНМО, 2016.

## Grading System

**Intermediate grade-2** = (1/3) test-1 + (1/3) colloquium-2 + (1/3) homework-2.

**Cumulative grade-3** = (3/10) test-1 + (3/10) colloquium-2 + (4/10) homework-3.

**Final grade-3** = min(10, (7/10) cumulative grade-3 + (3/10) final exam + (1/10) bonus points).

The number in a grade’s name is the number of the module when grading takes place. The grade homework-n is the normalized average grade for the homework in Modules from 1 to n. The Intermediate and Final grades are subject to rounding half up to an integer. All the other grades are reported with the greatest precision available.

Bonus point number is between 0 to 20. Such points may be given for a variety of auxiliary activities.