Elements of Discrete Mathematics

Lecturer:

Karel Klouda

Tutor:

Karel Klouda

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 BIK-ZMA, BIK-MLO and BIK-LIN.

Syllabus of lectures:

1. Sets, cardinality, countable sets, power set of a finite set and its cardinality. Power set of the set of natural numbers - uncountable set.

2. Exclusion and inclusion, its use to determine cardinality. „Pigeon-hole principle“, number of structures, i.e., number of maps, relations, trees (on finite structures).

3. Function estimates (factorial, binomial coefficients, ...). Relation, equivalence relation (examples of equivalence of connected/strongly connected components).

4. Relation matrix, relational databases. Mathematical induction as a tool for determining the number of finite objects.

5. Mathematical induction as a tool for proving algorithm correctness. Mathematical induction as a tool for solving recursive problems.

6. Structural induction. Runtime complexity of recursive algorithms - solving recursive equations with constant coefficients, homogeneous equations.

7. Solving non-homogeneous recursive equations with constant coefficients.

Syllabus of tutorials:

1. Cardinality calculations. Countability, uncountability. Inclusion and exclusion principle. Numbers of structures over finite sets. Asymptotic function behavior.

2. Relations and directed graphs. Basic proofs by induction. Application of proofs by induction in combinatorics. Application of proofs by induction in programming. Induction and recursive algorithms.

3. Uses of induction in formal language theory. Runtime complexity calculations. Solving linear recurrent equations. Reserved.

Study Objective:

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:

Time-table is not available yet

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans:

• Informtion Systems and Management (Presented in Czech) (compulsory course in the program)
• Information Technology (Presented in Czech) (compulsory course in the program)
• Computer engineering (Presented in Czech) (compulsory course in the program)
• Software engineering (Presented in Czech) (compulsory course in the program)
• Computer Science (Presented in Czech) (compulsory course in the program)
• Web and Multimedia (Presented in Czech) (compulsory course in the program)
• Informatics - Plan for Period Before Assignement of Specialization (Presented in Czech) (compulsory course in the program)