Discrete mathematicsSyllabus

Subject code

İF-17122y

Credits

5

Student workload

90

Semester

Autumn semester

Academic year

2

Subject type

Mandatory

Form of education

Full-time

Language of instruction

Azerbaijani

In-class hours

a) 30 saat - mühazirə b) 30 saat - seminar

Out-of-class hours

a) 45 saat - TSİ b) 45 saat - MRTSİ

The subject provides students with the formation of theoretical knowledge and analytical skills on the fundamental concepts and methods of discrete mathematics. within the subject, mathematical logic and logic of predicates, sets and relations, boolean algebra, basics of combinatorics and graph theory are studied. principles of correct formulation of mathematical concepts, logical analysis of reasoning and application of various proof methods are considered. bul functions, their separations, concepts of completeness and closure, multivalued functions, operations on graphs, paths and chains in graphs, connectivity, and other basic graph concepts are explored. within the scope of the subject, the application of discrete mathematics methods in computing technologies, automata theory, coding, information transfer and processing, development of algorithms and optimal software solutions is considered. special attention is also paid to the application of graph theory in the fields of network security and cyber forensics. the subject is aimed at developing students' abilities to analyze mathematical texts, make logical judgments, apply theoretical knowledge to solving practical problems, and model computing and information security problems using discrete structures.

• lecture
• presentations
• seminar

1. 1

   CLO 1: Must have knowledge of elementary functions of logic algebra and their properties. formulas, expression of functions with formulas, should be able to work with formulas of the same structure. must have knowledge of tautology and false formulas of identity, equivalence of formulas.

2. 2

   CLO 2: A dual function must have knowledge of the principle of duality. logic should be able to construct separation of algebraic functions by variables, perfect disjunctive forms, and perfect normal forms.

3. 3

   CLO 3: Must know bul functions and their simple applications in modeling digital circuits and cryptographic primitives.

4. 4

   CLO 4: Must know the essential elements of number theory used in public-key cryptography and their applications in appropriate cryptographic algorithms.

5. 5

   CLO 5: Must know the basic elements of finite squares and be aware of examples of their application in cryptographic algorithms.

6. 6

   CLO 6: Must know graph description methods and algorithms for finding shortest paths in weighted graphs and be able to apply them in network modeling (ospf).

1. 1.
   Elements of combinatorics. cartesian product. bul collectionsLecture

   The main concepts and elements of combinatorics, principles of counting in finite sets and methods of solving combinatorial problems are studied within the subject. the concept of the cartesian product, the construction of the cartesian product of sets and its main properties are reviewed. also, the concept of boolean sets, their structure and application in discrete mathematics is investigated.

2. 2.
   Concept of relationship. a binary relationship. equivalence relation.Lecture

   In the framework of the topic, the concept of relationship and the mathematical expression of relationships between the elements of sets are studied. the concept of binary relationship, its methods of presentation and main properties are considered. the equivalence relation, its properties of reflexivity, symmetry and transitivity, as well as the principles of dividing the set into equivalence classes are also investigated.

3. 3.
   Functions. composition of functions. the power of the multitudeLecture

   Within the scope of the topic, the concept of a function, the domain of a function and the set of values ​​are studied. the main types of functions and methods of their assignment, as well as the composition of functions and their main properties are considered. also, the concept of the power of a set, the principles of determining and comparing the power of finite and infinite sets are examined.

4. 4.
   Basic logical operations and their applicationsLecture

   The main concepts of mathematical logic and logical operations are studied within the subject. the essence, truth tables and basic properties of negation, conjunction, disjunction, implication and equivalence operations are reviewed. also, the construction, analysis and application of logical expressions in solving various practical issues are investigated.

5. 5.
   Formula concept. dual function. the principle of dualityLecture

   Within the subject, the concept of logical formula, the structure and main features of formulas are studied. the concept of binary function and its relationship with logical functions are considered. the principle of duality, the construction of dual forms of logical expressions and the application of this principle in boolean algebra are also investigated.

6. 6.
   3. detection and evaluation of conditions arising in the fh. 2Seminar

   Detection and evaluation of engineering conditions. determining the boundaries of spill and fire zones. problem solving.

7. 7.
   Separation of bul functions according to variables. perfect normal shapes. jegalkin polynomialLecture

   In the framework of the topic, methods of separation of bul functions according to variables and their main features are studied. the principles of expression of bul functions in the form of perfect disjunctive normal form (pmf) and perfect conjunctive normal form (mknf) are considered. also, the concept of jegalkin polynomial, presentation of boolean functions in algebraic normal form and methods of constructing jegalkin polynomial are investigated.

8. 8.
   Theory of groups. concepts of ring and squareLecture

   The basic concepts of algebraic structures are studied within the subject. the concept of group, the main properties of group operation, neutral and inverse elements are reviewed. the concepts of ring and square, their main properties and differences between them are explained. the role of group, ring, and field structures in discrete mathematics and other applied fields is also explored.

9. 9.
   Calculations by module. (z_m) class of residuesLecture

   Within the subject, the basic principles of calculations and the concept of comparison are studied by module. addition, subtraction, multiplication and exponentiation operations on integers are considered. the concept of the class of residues (z_m), the structure of the classes of residues and the principles of performing algebraic operations on them are investigated.

10. 10.
    Some numerical functions. RSA algorithmLecture

    The main numerical functions used in number theory and their properties are studied within the subject. basic concepts related to the euler function and modulus calculations are reviewed. mathematical basis of RSA algorithm, selection of prime numbers, generation of public and private keys, encryption and decryption processes are explained. also, the security of the RSA algorithm based on the problem of dividing large integers into primes is investigated.

11. 11.
    Finite periodic groups. primitive elements. discrete logarithm problemLecture

    In the framework of the topic, finite periodic groups, their structure and main properties are studied. group composition, element composition, and generator concepts are reviewed. the nature of primitive elements and their role in finite periodic groups are explained. also, the mathematical nature of the discrete logarithmization problem, the computational complexity of its solution, and its role in the security of cryptographic systems are investigated.

12. 12.
    Finite square. simple and extended finite squaresLecture

    The concept of finite field, its structure and main properties are studied within the subject. the principles of constructing simple finite squares and performing basic algebraic operations on their elements are considered. the structure of extended finite squares, their relationship with simple finite squares and their main properties are explained. also, the application of finite fields in coding, cryptography and other computational fields is investigated.

13. 13.
    Constructing finite squares. finite squares in the AES cipherLecture

    Within the framework of the topic, the principles of constructing finite squares and their application in cryptography are studied. the structure of simple and extended finite squares, the creation of extended squares based on irreducible polynomials, and the performance of algebraic operations on square elements are considered. the structure of the finite square (gf(2^8)) used in the AES cipher, the polynomial expression of bytes, and the addition and multiplication operations on the finite square are explained. also, the role of finite field operations in the basic transformations of the AES algorithm is investigated.

14. 14.
    Primitive polynomialsLecture

    Within the subject, the concept of primitive polynomial, its relation with finite fields and its main properties are studied. the differences between irreducible and primitive polynomials, the principles of defining primitive polynomials, and their role in the construction of finite fields are reviewed. also, the application of primitive polynomials in generating pseudorandom sequences, coding and cryptographic systems is investigated.

15. 15.
    Elements of graph theory. dijkstra's algorithmLecture

    The main concepts and elements of graph theory are studied within the subject. concepts of graph, vertex, thread, directed and undirected, weighted and unweighted graph, as well as path and connectivity in graphs are reviewed. the working principle of dijkstra's algorithm and its application in finding the shortest paths from a given starting vertex to other vertices in weighted graphs are explained. also, the main stages of the algorithm and application conditions are investigated.

1. Amirova r. a. discrete mathematics and mathematical logic.
2. Huseynov a.a., seyidov m.i., mammadov v.m. graph theory. teaching materials. baku-2018
3. Huseynov a.a. discrete mathematics and mathematical logic (parts i, ii). teaching aids. baku-2016.