This document summarizes the course content and structure for Discrete Mathematics at the National Research Ogarev Mordovia State University. The course is divided into 4 modules covering set theory, graph theory, algebraic structures, and coding theory. Students take exams and write 3 essays throughout the semester to assess their understanding of each module. Pedagogical methods include lectures, practice problems, subgroup work, computer programming assignments, and a final exam to evaluate students on a 100 point scale.

1. NATIONAL RESEARCH OGAREV MORDOVIA STATE UNIVERSITY
Course: Discrete Mathematics
Major profiles:
Informatics and computer science (ICS), Programming Engineering (PE)

2. CONTENT – 1
The discipline “Discrete Mathematics” (hereinafter referred to as DM) is studied
at the 2st (spring) semester of the 1st (2nd) year of tuition.
The content is divided into 4 blocks / modules:
• “Set theory and combinatorics”.
• “Graph theory”.
• “Algebraic structures”.
• “Coding theory”.
The volumes of the 1st and the 2nd modules (in academic hours) are
approximately the same, the 3rd and the 4th volumes are somewhat smaller.
At the end of studying the module 1, 2 and 3–4 every student must write and
defend an essay (will be discussed later). So there are 3 essays to be written
during the semester.
The studying of DM ends with exam that takes place in June.
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3. CONTENT – 2
Set theory and combinatorics
The definition of set. Universal and empty sets. Set operations and their
properties. Venn diagrams.
Cartesian product of sets. Binary relations. Application to relational databases.
Ordering and equivalence relationships. Cardinality of a set. Mappings. Ordered
and disordered sets. Sorting algorythms.
Problems of combinatorics. Principles of sum and of product. Permutations,
arrangements and combinations (with and without repeating elements). The
number of subsets in n-element set. Binomial theorem and its generalization.
The Pascal triangle.
Coverings and partitions. The cardinality of sets’ union. Stirling (of the 2nd kind)
and Bell numbers.
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4. CONTENT – 3
Graph theory
Graphs. adjacency and incidence. The ways of graph representation: adjacency
and incidence matrices, list of adjacency. Basic operations upon graphs.
Graph connectivity. Distances in graphs. Tree graphs and circuits. Cyclomatic
number. Euler (unicursal) and Hamilton graphs. Spanning trees. Algorithms of
Prim and Kruskal.
Depth-first search and breadth-first search. Modification of algorithms for
searching of graph’s connectivity components.
Transport networks. Dijkstra algorithm. The problems of maximum flow and
Ford-Fulkerson algorithm.
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5. CONTENT – 4
Algebraic structures
Operation upon the elements of set. Algebra as some structure (elements +
operations). Cayley table.
Sets with one operation. Groupoids and groups. Examples. Subgroup. Abel
(commutative) groups. The group of permutations. Cyclic groups.
Sets with two operations. Rings and fields.
The ring of integers. Divisibility, GCD and LCM. Euclidean algorithm due to
integers. Comparisons by integer module. Rings and fields of residues.
The ring of one-variable polynomials. The polynomial roots. Divisibility and GCD
of polynomials. Fundamental algebra theorem. Factorization of polynomials.
Horner’s rule. Euclidean algorithm due to polynomials.
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6. CONTENT – 5
Coding theory
Coding as mapping, its purposes and types. Uniquely decodable codes. Kraft-
McMillan theorem. Prefix codes.
Message redundancy and the problem of optimal coding. Example: Morse code.
Shannon-Fano and Huffman codes.
Elementary mistakes in messages: symbol drop-out, inserting of symbol and
substitution error. Noise combating codes. The code distance. Hamming code.
Coding for data compression. LZW algorithm. The idea of compressing data with
information losses (JPEG and MP3).
Coding for information security. Ciphers. Public key encryption. Diffie-Hellman
key exchange and RSA algorithm.
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7. Pedagogical METHODS – 1
Auditorial
• Lections – for all 1st (2nd)-year students of the profile.
• Practice / trainings – for each academic group separately (if there are many).
On practice / trainings students solve typical computational problems of
the discipline proposed by the teacher. The topics of practice / trainings
must be in harmony with the lections’ topics.
Subgroup (5– 6 students) work is done from time
to time at the trainings. It may be a collective
solving of a problem that allows “parallelization”
(e.g. searching for spanning tree using Prim and
Kruskal algorithms). Students enjoy this kind of
command work as it is a competition between
subgroups.
It’s quite effective, too, because students explain the learning
material to each other while fulfilling the collective task.
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8. Extraauditorial
• Essays / tests – mandatory
For DM essay is a sequence of theoretical questions and computational
problems that must be solved by a student. 3 essays must be written: in the
middle of March, April and May, respectively.
• Computer programming – additional
The PE and ICS-students are the future programmers, so they must be able
to realize some typical algorithms of DM using high-level programming language.
The student may choose the language as he/she wants, dialects of C and Pascal
are the common choices.
Pedagogical METHODS – 2
All the extraauditorial work is
controlled by the teacher.
Students must defend their
essays and programs.
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9. Pedagogical METHODS – 3
Example of computational problems in essay “Graph theory”
1. Two graphs are described by their matrices of adjacency. Check if the graphs
are isomorphic. Then make a picture of both graphs. (Each graph has 7 vertices.)
2. Two graphs are given in a picture. Check if the graphs are homeomorphic. (The
number of vertices is some like 10–12 and may not be same for both graphs.)
3. In one of graphs mentioned above (see #2) find the Euler subgraph with
maximal number of edges.
4. For a graph on the picture find diameter, centers, radius, articulation nodes and
bridges. Write down the incidence matrix for this graph.
5. 10-vertice graph is given by its matrix of adjacency. Find the way from the 3rd
vertex to the 10th vertex using depth-first search. Find the way from the 10th
vertex to the 1st vertex using breadth-first search. Don’t make a picture of the
graph while fulfilling the task.
…to be continued on the next slide
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10. … the beginning is on the previous slide
6. 10-vertice graph is given by its list of adjacency. Find all the connectivity
components of the graph by using depth-first search and breadth-first search. As
in the previous task, you should not make a picture of the graph.
7. Find the minimal spanning tree of weighted graph using algorithms of Prim and
Kruskal.
8. Weighted graph from #7 represents transport network. Find the minimal-cost
way from vertex 0 to vertex 10 using Dijkstra algorithm.
9. Find the maximal flow in a transport network with the help of Ford-Fulkerson
algorithm.
Pedagogical METHODS – 4
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11. Pedagogical METHODS – 5
Examples of programming tasks (module “Graph theory”)
(The ways of graph representation are defined by a programmer; the choice of
algorithm relies on a programmer, too).
1. Find the way from vertex A to vertex B in a connected graph (A and B are
chosen by user).
2. Find the minimal spanning tree in a weighted graph.
3. In a transport network find the
minimal-cost way from the source to
the sink.
4. Find out the maximal flow in the
transport network.
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12. ASSESSMENTS
Rating system is used: semester (3 essays + auditorial work) + exam
100 points is the possible maximum
• 86 – 100 – “Excellent” (approx. “A” or “B” in ECTS grading scale)
• 71 – 85 – “Good” (approx. “C” or “D” in ECTS grading scale)
• 51 – 70 – “Satisfactory” (approx. “D” or “E” in ECTS grading scale)
• 0 – 50 – “Bad” (approx. “Fx” or “F” in ECTS grading scale)
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Example of examination task
Theoretical questions
1. Binary relations and their properties. Ordering relationship.
2. Transport networks. Dijkstra algorythm.
Computational problem
Correct the substitution error in a message which is coded by (15, 11)
Hamming code, if there is any: 001010111010011.