Elements of Discrete Mathematics
Code  Completion  Credits  Range  Language 

BIZDM  Z,ZK  5  2+2  Czech 
 Lecturer:
 Daniel Dombek, Josef Kolář (guarantor)
 Tutor:
 Daniel Dombek, Josef Kolář (guarantor), Luděk Kleprlík, Petr Matyáš, Eva Pernecká, Jiřina Scholtzová, Jan Spěvák
 Supervisor:
 Department of Applied Mathematics
 Synopsis:

Students get both a mathematical sound background, but also practical calculation skills in the area of combinatorics, value estimation and formula approximation, tools for solving recurrent equations, and basics of graph theory.
 Requirements:

Students should have an adequate knowledge of basic notions of mathematics and mathematical logic as presented in previous subjects BIZMA, BIMLO and BILIN.
 Syllabus of lectures:

1. Sets, cardinality, countable sets, power set of a finite set and its cardinality.
2. Power set of the set of natural numbers  uncountable set.
3. Exclusion and inclusion, its use to determine cardinality.
4. „Pigeonhole principle“, number of structures, i.e., number of maps, relations, trees (on finite structures).
5. Function estimates (factorial, binomial coefficients, ...).
6. Relation, equivalence relation (examples of equivalence of connected/strongly connected components).
7. Relation matrix, relational databases.
8. Mathematical induction as a tool for determining the number of finite objects.
9. Mathematical induction as a tool for proving algorithm correctness.
10. Mathematical induction as a tool for solving recursive problems.
11. Structural induction.
12. Runtime complexity of recursive algorithms  solving recursive equations with constant coefficients, homogeneous equations.
13. Solving nonhomogeneous recursive equations with constant coefficients.
 Syllabus of tutorials:

1. Cardinality calculations.
2. Countability, uncountability.
3. Inclusion and exclusion principle.
4. Numbers of structures over finite sets.
5. Asymptotic function behavior.
6. Relations and directed graphs.
7. Basic proofs by induction.
8. Application of proofs by induction in combinatorics.
9. Application of proofs by induction in programming.
10. Induction and recursive algorithms.
11. Uses of induction in formal language theory.
12. Runtime complexity calculations.
13. Solving linear recurrent equations.
 Study Objective:
 Study materials:

1. Johnsonbaugh, R. Discrete Mathematics (4th Edition). Prentice Hall, 1998. ISBN 0130805505.
 Note:
 Further information:
 https://courses.fit.cvut.cz/BIZDM/
 Timetable for winter semester 2018/2019:

06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon Tue Fri Thu Fri  Timetable for summer semester 2018/2019:
 Timetable is not available yet
 The course is a part of the following study plans:

 Information Technology  Version for those who Enrolled in 2012 (in Czech) (compulsory course in the program)
 Informatics (Bachelor) Version for those who Enrolled in 2013 (in Czech) (compulsory course in the program)
 Web and Multimedia Version for those who Enrolled in 2013 (in Czech) (compulsory course in the program)
 Computer Science  Version for those who Enrolled in 2013 (in Czech) (compulsory course in the program)
 Information Technology  Version for those who Enrolled in 2013 (in Czech) (compulsory course in the program)
 Software Engineering Version for those who Enrolled in 2013 (in Czech) (compulsory course in the program)
 Information Systems and Management  Version for those who Enrolled in 2013 (in Czech) (compulsory course in the program)
 Informatics (Bachelor) Version for those who Enrolled in 2014 (in Czech) (compulsory course in the program)
 Web and Multimedia Version for those who Enrolled in 2014 (in Czech) (compulsory course in the program)
 Computer Science  Version for those who Enrolled in 2014 (in Czech) (compulsory course in the program)
 Information Technology  Version for those who Enrolled in 2014 (in Czech) (compulsory course in the program)
 Computer Engineering, Version for those who Enrolled in 2014, in Czech (compulsory course in the program)
 Software Engineering Version for those who Enrolled in 2014 (in Czech) (compulsory course in the program)
 Information Systems and Management  Version for those who Enrolled in 2014 (in Czech) (compulsory course in the program)
 Bc. Programme Informatics, in Czech, Version 2015, 2016, 2017 and 2018 (compulsory course in the program)
 Bc. Branch Security and Information Technology, in Czech, Version 2015, 2016, 2017 and 2018 (compulsory course in the program)
 Bc. Branch Computer Science, in Czech, Version 2015, 2016, 2017 and 2018 (compulsory course in the program)
 Bc. Branch Computer Engineering, in Czech, Version 2015, 2016, 2017 and 2018 (compulsory course in the program)
 Bachelor Branch Information Systems and Management, in Czech, Version 2015, 2016, 2017 and 2018 (compulsory course in the program)
 Bachelor Branch Knowledge Engineering, in Czech, Version 2015, 2016 and 2017 (compulsory course in the program)
 Bachelor Branch WSI, Specialization Software Engineering, in Czech, Version 2015 2016, 2017 and 2018 (compulsory course in the program)
 Bachelor Branch, Specialization Web Engineering, in Czech, Version 2015 2016, 2017 and 2018 (compulsory course in the program)
 Bachelor Branch WSI, Specialization Computer Grafics, in Czech, Version 2015, 2016, 2017 and 2018 (compulsory course in the program)
 Bachelor Branch Knowledge Engineering, in Czech, Version 2018 (compulsory course in the program)