Question

Элементы теории множеств

Answer 1

N - Множество натуральных чисел;

Z – множество целых чисел;

Q – множество рациональных чисел;

R - множество действительных чисел;

С - множество комплексных чисел.

Answer 2

Elements of Set Theory

Set theory is a branch of mathematical logic that deals with the study of sets, which are collections of distinct objects. It provides a foundation for various areas of mathematics and has applications in computer science, linguistics, and other fields.

Sets: In set theory, a set is a well-defined collection of distinct objects, called elements or members of the set. Sets are typically denoted by capital letters, and their elements are listed within curly braces. For example, if we have a set A containing the elements 1, 2, and 3, we can write it as A = {1, 2, 3}.

Operations on Sets: Set theory defines several operations that can be performed on sets. Some of the key operations include:

- Union: The union of two sets A and B, denoted by A ∪ B, is the set that contains all the elements that are in either A or B, or in both. - Intersection: The intersection of two sets A and B, denoted by A ∩ B, is the set that contains all the elements that are common to both A and B. - Complement: The complement of a set A, denoted by A', is the set that contains all the elements that are not in A but are in the universal set. - Difference: The difference between two sets A and B, denoted by A - B, is the set that contains all the elements that are in A but not in B.

Set Relations: Set theory also studies relations between sets. Some important relations include:

- Subset: A set A is said to be a subset of another set B if every element of A is also an element of B. This is denoted by A ⊆ B. - Proper Subset: A set A is said to be a proper subset of another set B if A is a subset of B, but there exists at least one element in B that is not in A. This is denoted by A ⊂ B. - Equality: Two sets A and B are said to be equal if they have exactly the same elements. This is denoted by A = B.

Cardinality: The cardinality of a set is the number of elements it contains. It is denoted by |A|, where A is the set. For example, if A = {1, 2, 3}, then |A| = 3.

These are some of the basic elements and concepts of set theory. Set theory provides a foundation for various mathematical concepts and is widely used in different areas of study.