470-2301/04 – Discrete Mathematics (DIM)
| Gurantor department | Department of Applied Mathematics | Credits | 5 |
| Subject guarantor | doc. Mgr. Petr Kovář, Ph.D. | Subject version guarantor | doc. Mgr. Petr Kovář, Ph.D. |
| Study level | undergraduate or graduate | | |
| | Study language | English |
| Year of introduction | 2019/2020 | Year of cancellation | |
| Intended for the faculties | FEI | Intended for study types | Bachelor |
Subject aims expressed by acquired skills and competences
The goals of this subject are to introduce basic terms and methods of discrete mathematics and graph theory and to teach students to use them to formulate real life problems using mathematical terminology and to solve these related practical problems.
The students should learn to
- comprehend and generalize given definitions,
- distinguish which theoretical approach is suitable for a particular practical problem,
- classify given objects based on given properties and summarize the results,
- summarize each chapter.
In class students should practice to
- state a real life problem in the terms of Discrete mathematics or Graph theory
- apply theoretical approaches in solving real life problem, choose proper methods,
- choose and compare various methods of solution and pick the most suitable,
- verify the validity of each partical method,
- reuse the same theoretical approach in similar problems.
Teaching methods
Lectures
Individual consultations
Tutorials
Project work
Summary
In this course the students learn the basic terms and concepts and basic constructions used in Discrete mathematics, especially in Combinatorics and Graph theory.
The word "discrete" refers to the opposite of "continuous". In this course we deal almost exclusively with finite sets and finite objects.
Compulsory literature:
Recommended literature:
K.H.Rosen, Discrete Mathematics and Its Applications - 6th ed., McGraw-Hill, New York NY, (2007), ISBN-10 0-07-288008-2.
P. Kovář: Lecture slides online http://homel.vsb.cz/~kov16/files/dim_kapitoly_all_en.pdf
P. Kovář, T. Kovářová: Solved exercises http://homel.vsb.cz/~kov16/files/dim_solved_exercises.pdf
Additional study materials
Way of continuous check of knowledge in the course of semester
Quizes during the semester.
Each week (starting with the third week) there will be short quizes at the beginning of each class (10 quizes, 2 or 3 points each). No books or notes are alowed to be used to write the quiz.
At the end of the semester eight best quizes will be counted, four of each type. Maximum is 20 points.
Topics of the quizes can be foud on the course webpage (webpage of the lecturer). The exercises on the quiz will be similar but not identical.
Project.
Topis will be available in the second half of the semester. A regular project consists of several problem, half in Discrete mathematics, half in Graph theory. The problems will be graded 0-2 or 0-3 point depending on difficulty, the total is 10 points.
There can be bonus projects available, each with two more difficult problems (Discrete mathematics +Graph theory), every student can work on the bonus project instead of the regular project. The problems will be graded 0/3/5 points, in the case of exceptionaly good solution 10 points.
Exam.
A student can come for the exam oly if he passed "zápočet". Main part of the exam is a written test for approximately 75 minutes. During the written part students can use one page A4 of hand-written notes. This legalized "cheat-sheet" may contain definitions, formulas, thorems, but NO SOLVED EXAMPLES!
Necessary conditions to pass "zápočet":
Get at least 10 point on quizes and project. The project has to be "accepted". A project can be accepted if it fulfils all formal requirement, i.e. title page, ID, etc. A project can be accepted even if it gains no points.
E-learning
Other requirements
No additional requirements are imposed on the student.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
Lectures and discussions:
Part I - Introduction to discrete mathematics.
The scope and goals of Discrete mathematics. Counting selections and arrangements, related problems. Pidgeon-hole principle.
Geometric and arithmetic progressions. Inclusion-exclusion principle, binomial theorem and its applications.
Discrete probability: tossing coins, random choice, shuffling cards. Probability space and random event.
Integers and mathematical induction, the concept of proofs. Proving the formulas for numbers of subsets, k-combinations, k-permutations.
Recurrence relations, their applications and examples.
Modular arithmetics.
Algorithmic aspects: practical implementation of sets and relations. Obtaining a full list of all possible subsets k-combinations, and k-permutations.
Part II - Introduction to graph theory
Graphs and relations. Subgraphs, isomorphism, degrees, implementation. Directed graphs.
Graph connectivity, algorithms for searching. Higher degrees of connectivity, edge-connectivity. Eulerian graphs and their applications.
Distance in graphs, Dijkstra's algorithm, graph metric and its enumeration.
Trees and their characterizations, rooted trees, tree isomorphism.
Spanning trees, MST problem. Basic algorithms for MST.
Planar embeddings of graphs, Euler's formula and its applications.
Graph coloring, bipartite graphs, recognition.
Network flows: formulation and application to real life problems. Ford-Fulkerson's algorithm for maximum flow. Further applications to matching and system of representatives.
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction