MA0002 Discrete Mathematics

Faculty of Education
Autumn 2023
Extent and Intensity
2/0/0. 4 credit(s). Type of Completion: zk (examination).
Teacher(s)
Mgr. Helena Durnová, Ph.D. (lecturer)
Guaranteed by
Mgr. Helena Durnová, Ph.D.
Department of Mathematics – Faculty of Education
Supplier department: Department of Mathematics – Faculty of Education
Timetable
Mon 7:00–8:50 učebna 35
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The course serves as propaedeutics for theoretical mathematical disciplines (algebra and mathematical analysis). The aim of the course is to prepare the students for the study of theoretical mathematical disciplines, namely algebra and mathematical analysis. During the lectures, students will learn about using symbolic language in concrete examples, in the homework assignment, they will learn these methods through solving a set of tasks. Students will thus learn in discrete forms the methods they will later use in mathematical analysis and algebra.
Learning outcomes
Successful graduates of the course will be prepared for the study of theoretical mathematical disciplines, i. e. algebra and mathematical analysis.
Syllabus
  • 1. Introduction to discrete mathematics. Basic combinatorial rules. 2. Factorial. Combinatorial numbers. Basic categories of combinatorics according to Jacob Bernoulli and their derivation. 3. Combintaorial tasks with numbers and with geometric shapes. 4. Binomial theorem nad arithmetic triangle. 5. Problem tasks in combinatorics. Combinatorial proof. 6. Recurrent formulas.Sequences in discrete mathematics. 7. Finite and partial sums. Proofs using mathematical induction. 8. Polynomials. Roots of polynomials. Dividing a polynomial by a polynomial. 9. Divisibility of polynomials. The greatest common divisor and the least common multiple of a polynomial. 10. Diophantine equations, divisibility and congruences. 11. Basic notions in graph theory. 12. Problems without a solution (especially in graph theory).
Literature
    recommended literature
  • SMULLYAN, Raymond M. Jak se jmenuje tahle knížka? Translated by Antonín Vrba - Hanuš Karlach. Vydání druhé, upravené,. Praha: Portál, 2015, 198 stran. ISBN 9788026208228. info
  • HERMAN, Jiří, Radan KUČERA and Jaromír ŠIMŠA. Metody řešení matematických úloh I. 3. vyd. Brno: Masarykova univerzita, 2011, 278 pp. ISBN 978-80-210-5636-7. info
  • HERMAN, Jiří, Radan KUČERA and Jaromír ŠIMŠA. Seminář ze středoškolské matematiky. 1. dotisk 2., přeprac. vyd. Brno: Masarykova univerzita, 2007, 51 s. ISBN 978-80-210-3528-7. info
  • HERMAN, Jiří, Radan KUČERA and Jaromír ŠIMŠA. Counting and Configurations: Problems in Combinatorics, Arithmetic, and Geometry. 1st ed. New York: Springer-Verlag, 2003, 410 pp. Canadian Mathematical Society Books in Math., 12. ISBN 0-387-95552-6. info
  • HERMAN, Jiří, Jaromír ŠIMŠA and AT AL. Sbírka testových úloh k maturitě z matematiky (Testing problems for high-school leaving exams in mathematics). Praha: Prometheus, 2002, 279 pp. ISBN 80-7196-249-X. info
  • HERMAN, Jiří, Radan KUČERA and Jaromír ŠIMŠA. Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory. 1st ed. New York: Springer-Verlag, 2000, 355 pp. Canadian Mathematical Society Books in Math., 1. ISBN 0-387-98942-0. info
  • HERMAN, Jiří, Radan KUČERA and Jaromír ŠIMŠA. Metody řešení matematických úloh II (Methods how to solve mathematics exercises II). Brno: Masarykova univerzita Brno, 1997, 355 pp. ISBN 80-210-1630-2. info
  • HERMAN, Jiří, Radan KUČERA and Jaromír ŠIMŠA. Metody řešení matematických úloh I. 2., přeprac. vyd. Brno: Masarykova univerzita, 1996, 278 s. ISBN 80-210-1202-1. info
  • VRBA, Antonín. Grafy : pro III. ročník tříd gymnázií se zaměřením na matematiku, na matematiku a fyziku a pro seminář a cvičení z matematiky ve IV. ročníku gymnázií. 1. vyd. Praha: Státní pedagogické nakladatelství, 1989, 75 s. info
  • VRBA, Antonín. Kombinatorika. 1. vyd. Praha: Mladá fronta, 1980, 130 s. URL info
  • VILENKIN, Naum Jakovlevič. Kombinatorika. Vyd. 1. Praha: SNTL - Nakladatelství technické literatury, 1977, 298 s. URL info
  • VRBA, Antonín. Princip matematické indukce. 1. vyd. Praha: Mladá fronta, 1977, 138 s. URL info
Teaching methods
Theoretical lectures and homework assignments: students will turn in a set of solved tasks for each topic (including commentaries).
Assessment methods
Written and oral exam.
Language of instruction
Czech
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
Due to the situation in FALL 2020, homeworks will not be turned in regularly, but please, bear in mind that they will take you quite some time. Bring the exercise book with your calculations to the exam. The layout and careful handwriting is irrelevant and so are the actual results, since master solutions will be made available to you. -------------------------- Homeworks must be turned in in the course of the semester, at the latest two weeks after they have been assigned. A handwritten hard copy is required. The sheets will be returned to the student after they have been checked. DO NOT send the homework through e-mail. Each sheet must contain the name, surname and identification number of the student. Possibilities to turn the homework in: - na přednášce - do schránky Katedry matematiky, s označením vyučující (Durnová) - na vrátnici Poříčí 31 (NIKOLIV Poříčí 9) s označením katedry a vyučující (Katedra matematiky, Durnová) V případě dlouhodobé nemoci nebo jiných závažných okolností je možno se domluvit jinak.
The course is also listed under the following terms Autumn 2017, Autumn 2018, Autumn 2019, autumn 2020, Autumn 2021, Autumn 2022, Autumn 2024.
  • Enrolment Statistics (Autumn 2023, recent)
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