Discrete MathematicsTopics

This topic introduces the fundamental concepts and elements of combinatorics, counting principles for finite sets, and methods for solving combinatorial problems. the concept of the cartesian product, construction of the cartesian product of sets, and its fundamental properties are examined. the concept of boolean sets, their structure, and applications in discrete mathematics are also studied.

This topic examines the concept of a relation and the mathematical representation of relationships between elements of sets. the concept of a binary relation, its methods of representation, and fundamental properties are considered. equivalence relations, their properties of reflexivity, symmetry, and transitivity, as well as the principles of partitioning a set into equivalence classes, are also studied.

This topic introduces the concept of a function, including its domain and range. the main types and methods of representing functions, as well as the construction and fundamental properties of function composition, are examined. the concept of the cardinality of a set, methods for determining and comparing the cardinalities of finite and infinite sets, are also studied.

This topic examines the fundamental concepts and logical operations of mathematical logic. the meaning, truth tables, and fundamental properties of negation, conjunction, disjunction, implication, and equivalence are studied. the construction and analysis of logical expressions and their applications to solving various practical problems are also considered.

This topic introduces the concept of a logical formula, its structure, and fundamental properties. the concept of a dual function and its relationship with logical functions are examined. the principle of duality, construction of dual forms of logical expressions, and applications of this principle in boolean algebra are also studied.

Identification and assessment of the engineering situation. determination of the boundaries of destruction and fire zones. problem solving.

This topic examines methods for decomposing boolean functions with respect to variables and their fundamental properties. the principles of representing boolean functions in perfect disjunctive normal form (pdnf) and perfect conjunctive normal form (pcnf) are studied. the concept of the zhegalkin polynomial, algebraic normal representation of boolean functions, and methods for constructing zhegalkin polynomials are also examined.

This topic introduces the fundamental concepts of algebraic structures. the concept of a group, fundamental properties of group operations, identity and inverse elements are examined. the concepts of rings and fields, their fundamental properties, and the differences between them are explained. the role of group, ring, and field structures in discrete mathematics and other applied fields is also considered.

This topic examines the fundamental principles of modular arithmetic and the concept of congruence. addition, subtraction, multiplication, and exponentiation modulo an integer are considered. the concept of the residue class (zₘ), the structure of residue classes, and the principles of performing algebraic operations on them are also studied.

This topic examines fundamental arithmetic functions used in number theory and their properties. euler’s totient function and fundamental concepts related to modular arithmetic are considered.

This topic examines finite cyclic groups, their structure, and fundamental properties. the order of a group, order of an element, and the concept of a generator are considered. the concept of primitive elements and their role in finite cyclic groups are explained. the mathematical nature of the discrete logarithm problem, its computational complexity, and its role in the security of cryptographic systems are also studied.

This topic examines the concept of a finite field, its structure, and fundamental properties. the construction of prime finite fields and the principles of performing basic algebraic operations on their elements are considered. the structure of extension finite fields, their relationship with prime finite fields, and their fundamental properties are explained. applications of finite fields in coding, cryptography, and other computational fields are also examined.

This topic examines the principles of constructing finite fields and their applications in cryptography. the structure of prime and extension finite fields, the construction of extension fields using irreducible polynomials, and algebraic operations on field elements are considered.

This topic introduces the concept of primitive polynomials, their relationship with finite fields, and their fundamental properties. the differences between irreducible and primitive polynomials, principles for identifying primitive polynomials, and their role in the construction of finite fields are considered. applications of primitive polynomials in generating pseudorandom sequences, coding, and cryptographic systems are also studied.

This topic examines the fundamental concepts and elements of graph theory. the concepts of graphs, vertices, edges, directed and undirected graphs, weighted and unweighted graphs, as well as paths and connectivity in graphs, are considered. the operating principle of dijkstra’s algorithm and its application to finding the shortest paths from a given source vertex to other vertices in weighted graphs are explained. the main stages of the algorithm and its conditions of application are also examined.