1. Sets 1.1. Basic concepts of set theory

One of the basic, initial concepts of mathematics are the concepts of the set and its elements. By set we mean the totality of definite, different from each other, but of the same type objects, called elements of sets, subject to counting. Thus, the set consists of elements. It is customary to designate the sets in uppercase Latin letters ( A , B , C , ..., Z ), and the elements of the sets - in lower case letters ( a , b , c , ..., z ), numbers, and also identification expressions. Examples of sets are: the set of natural numbers, the number of students in a group, the number of groups in the faculty, etc.

The belonging of an element a to the set M is denoted by  (a M) ; non-belonging of an element a to the set M is denoted by  (a M) .

A set A is called a subset of B (A  B) if every element of A is an element of B. It is said that B contains or covers the set A. The sets A and B are equal (A = B) if their elements are the same, otherwise: A  B and B  A. The expression (A B) is a negation of the previous one. If A  B and B  A , A is often called a proper, strict or true subset of B (A  B) , The  sign is called the strict inclusion sign.

Sets can be finite, i.e. sets with a finite number of elements, and infinite. Given the practical significance of the course and its use to study the basics of the synthesis of finite automata, we will consider finite sets . The number of elements in a finite set M is called the power and is denoted by | M | . If the set contains no elements, then it is called empty and is denoted by . It is considered that the empty set is a subset of any set.

A set can be specified by an enumeration (a list of elements), a generating procedure, or a description of the characteristic properties that its elements must possess. The list can be set only finite sets. The list is enclosed in braces. For example, A = {a, b, d, h} means that the set A consists of four elements a , b , d and h .

The generating procedure describes a method for obtaining elements of a set of objects already received. An example is M - the set of all numbers of the form π / 2 ± kπ , where k  N ( N is the set of natural numbers).

The task of a set of a description of its properties, perhaps the most common. For example, the set of all even numbers from 0 to 100. When the property of elements can be described briefly by the expression P (x) , what means - the element x has the property P , then the set is given by the expression M = { x | P (x) }, which reads like this: M is a set of x , having the property of r . For example, M = { x | x = π / 2 ± kπ , where k N }. The description of the properties should be made with the requirement of accuracy and unambiguity.