Elements of Discrete Mathematics

Lecturer:

Josef Kolář (gar.)

Tutor:

Josef Kolář (gar.)

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, and tools for solving recurrent equations.

Requirements:

Fundamentals of calculus and algorithmics.

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:

The aim of the module is to learn students methods of combinatoric, asymptotic mathematic and mathematical induction as a basic tool for proofing correctness or deriving complexity of algorithms.

Study materials:

1. Johnsonbaugh, R. ''Discrete Mathematics (4th Edition)''. Prentice Hall, 1998. ISBN 0130805505.

2. Rosen, K. H. ''Discrete Mathematics and its Applications''. McGraw-Hill, 1998. ISBN 0072899050.

3. Cormen, T. H., Leiserson, C. E., Rivest, R. L. ''Introduction to Algorithms''. The MIT Press, 2001. ISBN 0262032937.

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
Tue	roomTH:A-1029 Kolář J. 07:30–09:00 (lecture parallel1) DejviceroomTH:A-1029 Kolář J. 09:15–10:45 (lecture parallel1 parallel nr.101) Dejvice
Fri
Thu
Fri

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans:

• Information Systems and Management (Presented in English) (compulsory course in the program)
• Information Technologies (Presented in English) (compulsory course in the program)
• Computer Engineering (Presented in English) (compulsory course in the program)
• Software Engineering (Presented in English) (compulsory course in the program)
• Computer Science (Presented in English) (compulsory course in the program)
• Web and Multimedia (Presented in English) (compulsory course in the program)
• Informatics (Presented in English) (compulsory course in the program)