Quite often in mathematics there are a number of difficulties and questions, and many of the answers are not always clear. No exception was such a topic as the cardinality of sets. In fact, this is nothing more than a numerical expression of the number of objects. In a general sense, a set is an axiom; it has no definition. It is based on any objects, or rather their set, which can be empty, finite or infinite. In addition, it contains integers or natural numbers, matrices, sequences, segments and lines.
About existing variables
A null or empty set with no intrinsic value is considered a cardinal element, as it is a subset. The collection of all subsets of a non-empty set S is a set of sets. Thus, the power set of a given set is considered to be many, conceivable, but single. This set is called the set of powers of S and is denoted by P(S). If S contains N elements, then P(S) contains 2^n subsets, since a subset of P(S) is either ∅ or a subset containing r elements from S, r=1, 2, 3, … Composed of everything infiniteset M is called a power quantity and is symbolically denoted by P (M).
Elements of set theory
This field of knowledge was developed by George Cantor (1845-1918). Today it is used in almost all branches of mathematics and serves as its fundamental part. In set theory, elements are represented in the form of a list and are given by types (empty set, singleton, finite and infinite sets, equal and equivalent, universal), union, intersection, difference, and addition of numbers. In everyday life, we often talk about a collection of objects such as a bunch of keys, a flock of birds, a pack of cards, etc. In math grade 5 and beyond, there are natural, integer, prime, and composite numbers.
The following sets can be considered:
- natural numbers;
- letters of the alphabet;
- primary odds;
- triangles with different sides.
It can be seen that these specified examples are well-defined sets of objects. Consider a few more examples:
- five most famous scientists in the world;
- seven beautiful girls in society;
- three best surgeons.
These cardinality examples are not well-defined collections of objects, because the criteria for "most famous", "most beautiful", "best" varies from person to person.
Sets
This value is a well-defined number of different objects. Assuming that:
- wordset is a synonym, aggregate, class and contains elements;
- objects, members are equal terms;
- sets are usually denoted by capital letters A, B, C;
- set elements are represented by small letters a, b, c.
If "a" is an element of the set A, then it is said that "a" belongs to A. Let's denote the phrase "belongs" with the Greek character "∈" (epsilon). Thus, it turns out that a ∈ A. If 'b' is an element that does not belong to A, this is represented as b ∉ A. Some important sets used in grade 5 mathematics are represented using the three following methods:
- applications;
- registries or tabular;
- rule for creating a formation.
On closer examination, the application form is based on the following. In this case, a clear description of the elements of the set is given. They are all enclosed in curly braces. For example:
- set of odd numbers less than 7 - written as {less than 7};
- a set of numbers greater than 30 and less than 55;
- number of students in a class that weigh more than the teacher.
In the registry (table) form, the elements of a set are listed within a pair of brackets {} and separated by commas. For example:
- Let N denote the set of the first five natural numbers. Therefore, N=→ register form
- Set of all vowels of the English alphabet. Hence V={a, e, i, o, u, y} → register form
- The set of all odd numbers is less than 9. Therefore, X={1, 3, 5, 7} → formregistry
- Set of all letters in the word "Math". Therefore, Z={M, A, T, H, E, I, C, S} → Registry Form
- W is the set of the last four months of the year. Therefore, W={September, October, November, December} → registry.
Note that the order in which the elements are listed does not matter, but they must not be repeated. An established form of construction, in a given case, a rule, formula or operator is written in a pair of brackets so that the set is correctly defined. In the set builder form, all elements must have the same property to become a member of the value in question.
In this form of set representation, an element of the set is described with the character "x" or any other variable followed by a colon (":" or "|" is used to indicate). For example, let P be the set of countable numbers greater than 12. P in the set-builder form is written as - {countable number and greater than 12}. It will read in a certain way. That is, "P is a set of x elements such that x is countable and greater than 12."
Solved example using three set representation methods: number of integers between -2 and 3. Below are examples of different types of sets:
- An empty or null set that does not contain any element and is denoted by the symbol ∅ and is read as phi. In list form, ∅ is written {}. The finite set is empty, since the number of elements is 0. For example, the set of integer values is less than 0.
- Obviously there shouldn't be <0. Therefore, thisempty set.
- A set containing only one variable is called a singleton set. Is neither simple nor compound.
Finite set
A set containing a certain number of elements is called a finite or infinite set. Empty refers to the first. For example, a set of all the colors in the rainbow.
Infinity is a set. The elements in it cannot be enumerated. That is, containing similar variables is called an infinite set. Examples:
- power of the set of all points in the plane;
- set of all prime numbers.
But you should understand that all cardinalities of the union of a set cannot be expressed in the form of a list. For example, real numbers, since their elements do not correspond to any particular pattern.
The cardinal number of a set is the number of different elements in a given quantity A. It is denoted n (A).
For example:
- A {x: x ∈ N, x <5}. A={1, 2, 3, 4}. Therefore, n (A)=4.
- B=set of letters in the word ALGEBRA.
Equivalent sets for set comparison
Two cardinalities of a set A and B are such if their cardinal number is the same. The symbol for the equivalent set is "↔". For example: A ↔ B.
Equal sets: two cardinalities of sets A and B if they contain the same elements. Each coefficient from A is a variable from B, and each of B is the specified value of A. Therefore, A=B. The different types of cardinality unions and their definitions are explained using the examples provided.
Essence of finiteness and infinity
What are the differences between the cardinality of a finite set and an infinite set?
The first value has the following name if it is either empty or has a finite number of elements. In a finite set, a variable can be specified if it has a limited count. For example, using the natural number 1, 2, 3. And the listing process ends at some N. The number of different elements counted in the finite set S is denoted by n (S). It is also called order or cardinal. Symbolically denoted according to the standard principle. Thus, if the set S is the Russian alphabet, then it contains 33 elements. It is also important to remember that an element does not occur more than once in a set.
Infinite in the set
A set is called infinite if the elements cannot be enumerated. If it has an unbounded (that is, uncountable) natural number 1, 2, 3, 4 for any n. A set that is not finite is called infinite. We can now discuss examples of the numerical values under consideration. End value options:
- Let Q={natural numbers less than 25}. Then Q is a finite set and n (P)=24.
- Let R={integers between 5 and 45}. Then R is a finite set and n (R)=38.
- Let S={numbers modulo 9}. Then S={-9, 9} is a finite set and n (S)=2.
- Set of all people.
- Number of all birds.
Infinite examples:
- number of existing points on the plane;
- number of all points in the line segment;
- the set of positive integers divisible by 3 is infinite;
- all whole and natural numbers.
Thus, from the above reasoning, it is clear how to distinguish between finite and infinite sets.
Power of the continuum set
If we compare the set and other existing values, then an addition is attached to the set. If ξ is universal and A is a subset of ξ, then the complement of A is the number of all elements of ξ that are not elements of A. Symbolically, the complement of A with respect to ξ is A'. For example, 2, 4, 5, 6 are the only elements of ξ that do not belong to A. Therefore, A'={2, 4, 5, 6}
A set with cardinality continuum has the following features:
- complement of the universal quantity is the empty value in question;
- this null set variable is universal;
- amount and its complement are disjoint.
For example:
- Let the number of natural numbers be a universal set and A be even. Then A '{x: x is an odd set with the same digits}.
- Let ξ=set of letters in the alphabet. A=set of consonants. Then A '=number of vowels.
- The complement to the universal set is the empty quantity. Can be denoted by ξ. Then ξ '=The set of those elements that are not included in ξ. The empty set φ is written and denoted. Therefore ξ=φ. Thus, the complement to the universal set is empty.
In mathematics, "continuum" is sometimes used to represent a real line. And more generally, to describe similar objects:
- continuum (in set theory) - real line or corresponding cardinal number;
- linear - any ordered set that shares certain properties of a real line;
- continuum (in topology) - non-empty compact connected metric space (sometimes Hausdorff);
- the hypothesis that no infinite sets are greater than integers but smaller than real numbers;
- the power of the continuum is a cardinal number representing the size of the set of real numbers.
Essentially, a continuum (measurement), theories or models that explain gradual transitions from one state to another without any abrupt change.
Problems of union and intersection
It is known that the intersection of two or more sets is the number containing all the elements that are common in these values. Word tasks on sets are solved to get basic ideas about how to use the union and intersection properties of sets. Solved the main problems of words onsets look like this:
Let A and B be two finite sets. They are such that n (A)=20, n (B)=28 and n (A ∪ B)=36, find n (A ∩ B)
Relationship in sets using Venn diagram:
- The union of two sets can be represented by a shaded area representing A ∪ B. A ∪ B when A and B are disjoint sets.
- The intersection of two sets can be represented by a Venn diagram. With shaded area representing A ∩ B.
- The difference between the two sets can be represented by Venn diagrams. With a shaded area representing A - B.
- Relationship between three sets using a Venn diagram. If ξ represents a universal quantity, then A, B, C are three subsets. Here all three sets are overlapping.
Summarizing set information
The cardinality of a set is defined as the total number of individual elements in the set. And the last specified value is described as the number of all subsets. When studying such issues, methods, methods and solutions are required. So, for the cardinality of a set, the following examples can serve as:
Let A={0, 1, 2, 3}| |=4, where | A | represents the cardinality of set A.
Now you can find your power pack. It's pretty simple too. As already said, the power set is set from all subsets of a given number. So one should basically define all the variables, elements and other values of A,which are {}, {0}, {1}, {2}, {3}, {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, { 2, 3}, {0, 1, 2}, {0, 1, 3}, {1, 2, 3}, {0, 2, 3}, {0, 1, 2, 3}.
Now power figure out P={{}, {0}, {1}, {2}, {3}, {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 3}, {0, 1, 2}, {0, 1, 3}, {1, 2, 3}, {0, 2, 3}, {0, 1, 2, 3}} which has 16 elements. Thus, the cardinality of the set A=16. Obviously, this is a tedious and cumbersome method for solving this problem. However, there is a simple formula by which, directly, you can know the number of elements in the power set of a given number. | P |=2 ^ N, where N is the number of elements in some A. This formula can be obtained using simple combinatorics. So the question is 2^11 since the number of elements in set A is 11.
So, a set is any numerically expressed quantity, which can be any possible object. For example, cars, people, numbers. In a mathematical sense, this concept is broader and more generalized. If at the initial stages the numbers and options for their solution are sorted out, then in the middle and higher stages the conditions and tasks are complicated. In fact, the cardinality of the union of a set is determined by the belonging of an object to a group. That is, one element belongs to a class, but has one or more variables.
