Elements of Discrete Mathematics

Lecturer:

Karel Klouda, Josef Kolář (gar.)

Tutor:

Karel Klouda, Josef Kolář (gar.), Peter Franek, Petr Matyáš, Štěpán Starosta

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 BI-ZMA, BI-MLO and BI-LIN.

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. „Pigeon-hole 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 non-homogeneous 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:

Time-table for winter semester 2011/2012:

	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	roomTK:BS Kolář J. 09:15–10:45 (lecture parallel1) Dejvice NTK Ballingův sálroomT9:301 Starosta Š. 14:30–16:00 (lecture parallel1 parallel nr.103) Dejvice NBFIT učebnaroomT9:301 Starosta Š. 16:15–17:45 (lecture parallel1 parallel nr.104) Dejvice NBFIT učebnaroomT9:301 18:00–19:30 (lecture parallel1 parallel nr.110) Dejvice NBFIT učebna roomTK:BS Klouda K. 14:30–16:00 (lecture parallel2) Dejvice NTK Ballingův sál
Tue	roomT9:344 07:30–09:00 (lecture parallel1 parallel nr.101) Dejvice NBFIT ucebnaroomT9:344 Matyáš P. 09:15–10:45 (lecture parallel1 parallel nr.102) Dejvice NBFIT ucebnaroomT9:344 Matyáš P. 11:00–12:30 (lecture parallel2 parallel nr.211) Dejvice NBFIT ucebnaroomT9:344 Matyáš P. 16:15–17:45 (lecture parallel2 parallel nr.206) Dejvice NBFIT ucebnaroomT9:344 Matyáš P. 18:00–19:30 (lecture parallel2 parallel nr.207) Dejvice NBFIT ucebna
Fri	roomT9:344 Franek P. 11:00–12:30 (lecture parallel2 parallel nr.203) Dejvice NBFIT ucebnaroomT9:344 Franek P. 12:45–14:15 (lecture parallel1 parallel nr.105) Dejvice NBFIT ucebnaroomT9:344 Franek P. 14:30–16:00 (lecture parallel2 parallel nr.205) Dejvice NBFIT ucebna
Thu	roomT9:344 07:30–09:00 (lecture parallel2 parallel nr.201) Dejvice NBFIT ucebnaroomT9:344 Starosta Š. 09:15–10:45 (lecture parallel2 parallel nr.202) Dejvice NBFIT ucebnaroomT9:344 Starosta Š. 11:00–12:30 (lecture parallel2 parallel nr.204) Dejvice NBFIT ucebnaroomT9:344 Starosta Š. 12:45–14:15 (lecture parallel1 parallel nr.106) Dejvice NBFIT ucebnaroomT9:344 Klouda K. 14:30–16:00 (lecture parallel1 parallel nr.107) Dejvice NBFIT ucebnaroomT9:344 Franek P. 16:15–17:45 (lecture parallel1 parallel nr.108) Dejvice NBFIT ucebnaroomT9:344 Franek P. 18:00–19:30 (lecture parallel1 parallel nr.109) Dejvice NBFIT ucebna
Fri	roomT9:344 07:30–09:00 (lecture parallel2 parallel nr.208) Dejvice NBFIT ucebnaroomT9:344 Klouda K. 09:15–10:45 (lecture parallel2 parallel nr.209) Dejvice NBFIT ucebnaroomT9:344 Klouda K. 11:00–12:30 (lecture parallel2 parallel nr.210) Dejvice NBFIT ucebna

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans:

• Computer Science, Version for Students who Enrolled in 2009 and 2010, Presented in Czech (compulsory course in the program)
• Computer engineering, Version for Students who Enrolled in 2009 and 2010, in Czech (compulsory course in the program)
• Software Engineering, Version for Students who Enrolled in 2009 and 2010, in Czech (compulsory course in the program)
• Web and Multimediac, Version for Students who Enrolled in 2009 and 2010, Presented in Czech (compulsory course in the program)
• Information Systems and Management, Version for Students who Enrolled in 2009 and 2010, in Czech (compulsory course in the program)
• Information Technology, Version for Students who Enrolled in 2009 and 2010, Presented in Czech (compulsory course in the program)
• Informatics, Version for Students who Enrolled in 2009 and 2010, Presented in Czech (compulsory course in the program)
• Informatics (Bachelor)- Version for those who Enrolled in 2011 and 2012 (in Czech) (compulsory course in the program)
• Information Systems and Management - Version for those who Enrolled in 2011 and 2012 (in Czech) (compulsory course in the program)
• Information Technology- Version for those who Enrolled in 2011 and 2012 (in Czech) (compulsory course in the program)
• Computer Engineering, Version for those who Enrolled in 2011 and 2012, in Czech (compulsory course in the program)
• Software Engineering- Version for those who Enrolled in 2011 and 2012 (in Czech) (compulsory course in the program)
• Computer Science - Version for those who Enrolled in 2011 and 2012 (in Czech) (compulsory course in the program)
• Web and Multimedia- Version for those who Enrolled in 2011 and 2012 (in Czech) (compulsory course in the program)