ČESKÉ VYSOKÉ UČENÍ TECHNICKÉ V PRAZE
STUDIJNÍ PLÁNY
2018/2019

# Elements of Discrete Mathematics

Kód Zakončení Kredity Rozsah Jazyk výuky
BIE-ZDM Z,ZK 5 2+2
Přednášející:
Josef Kolář (gar.)
Cvičící:
Josef Kolář (gar.)
Předmět zajišťuje:
katedra aplikované matematiky
Anotace:

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.

Fundamentals of calculus and algorithmics.

Osnova přednášek:

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.

Osnova cvičení:

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.

Cíle studia:

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.

Studijní materiály:

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.

Poznámka:

Informace o předmětu a výukové materiály naleznete na https://courses.fit.cvut.cz/BIE-ZDM/