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Explanatory Notes
Instructions
Web page:
https://moodle.fel.cvut.cz/courses/A8B01DMG
Anotation:
The course introduces basic notions from discrete mathematics directed to those topics useful for electrical engineering studies. The content of the course covers: infinite sets with emphasis to cardianlity of sets, binary relations with emphasis to equivalence relations and partial ordes'; integers, relation modulo n'; basic algebraic structures (includin finite fields of characteristic 2). Furher the course contains basic notions and their applications from graph theory.
Study targets:
The aim of the course is to introduce to students basic notions of discrete mathematics that will be used in their studies.
Content:
| 1. | | Sets. Cardinality of sets. |
| 2. | | Binary relalations, equivalence relation, partial order. |
| 3. | | Integers, Eclid's algorithm and Bezout's theorem |
| 4. | | Relation modulo n, rezidual classes and operations with them |
| 5. | | Binary operations, semigroups, groups. |
| 6. | | Sets with two binary operations, Boolean algebras. |
| 7. | | Rings of rezidual classes, finite fieldst of rezidual classes over a prime, polynomials |
over them.
| 8. | | Galois fields GF(2^k). |
| 9. | | Homomorfisms of algebraic structures. |
| 10. | | Undirected graphs, directed graphs, trees and spanning trees. |
| 11. | | Strongly connected and acyclic graphs, topological sort |
| 12. | | Combinatorics. |
| 13. | | Asymptotic growth of functions. |
Course outlines:
| 1. | | Sets. Cardinality of sets. |
| 2. | | Binary relalations, equivalence relation, partial order. |
| 3. | | Integers, Eclid's algorithm and Bezout's theorem |
| 4. | | Relation modulo n, rezidual classes and operations with them |
| 5. | | Binary operations, semigroups, groups. |
| 6. | | Sets with two binary operations, Boolean algebras. |
| 7. | | Rings of rezidual classes, finite fieldst of rezidual classes over a prime, polynomials |
over them.
| 8. | | Galois fields GF(2^k). |
| 9. | | Homomorfisms of algebraic structures. |
| 10. | | Undirected graphs, directed graphs, trees and spanning trees. |
| 11. | | Strongly connected and acyclic graphs, topological sort |
| 12. | | Combinatorics. |
| 13. | | Asymptotic growth of functions. |
Exercises outline:
| 1. | | Sets. Cardinality of sets. |
| 2. | | Binary relalations, equivalence relation, partial order. |
| 3. | | Integers, Eclid's algorithm and Bezout's theorem |
| 4. | | Relation modulo n, rezidual classes and operations with them |
| 5. | | Binary operations, semigroups, groups. |
| 6. | | Sets with two binary operations, Boolean algebras. |
| 7. | | Rings of rezidual classes, finite fieldst of rezidual classes over a prime, polynomials |
over them.
| 8. | | Galois fields GF(2^k). |
| 9. | | Homomorfisms of algebraic structures. |
| 10. | | Undirected graphs, directed graphs, trees and spanning trees. |
| 11. | | Strongly connected and acyclic graphs, topological sort |
| 12. | | Combinatorics. |
| 13. | | Asymptotic growth of functions. |
Literature:
| 1. | | Lindsay N. Childs: A Concrete Introduction to Higher Algebra, Springer; 3rd edition (November 26, 2008), |
ISBN-10: 0387745270
| 2. | | Jiří Demel: Grafy a jejich aplikace, Academia; 2002, ISBN 80-200-0990-6 |
| 3. | | Richard Johnsonbaugh: Discrete Mathematics, Prentice Hall, 4th edition (1997), ISBN 0-13-518242-5 |
Requirements:
None.
Keywords:
Sets, cardinality of sets, binary relations, equivalence relations, partial order, relation modulo n, rezidual classes and operations with them, semigroups, groups, finite fields, Boolean algebras, directed and undirected graphs, trees, spanning trees, strongly connected graphs, asymptotic growth of functions.
Subject is included into these academic programs:
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Proposal and Realization: I. Halaška (K336), J. Novák (K336) |