Quite often, a number of difficulties and questions arise in mathematical science, and many answers are not always clarified. No exception was such a topic as the power 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 zero or empty set that does not have its own value is considered an element of power, since this is a subset. The collection of all subsets of a nonempty set S is a set of sets. Thus, the set of power of a given set is considered to be many, conceivable, but single. This set is called the set of degrees S and is denoted by P (S). If S contains N elements, then P (S) contains 2 ^ n subsets, since the subset P (S) is either ∅ or a subset containing r elements from S, r = 1, 2, 3, ... Composed from of the entire infinite set 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 years of life). Today it is used in almost all branches of mathematics and serves as its fundamental part. In set theory, elements are presented 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 complement of numbers. In everyday life, it is often said about a collection of objects such as a bunch of keys, a flock of birds, a pack of cards, etc. In mathematics of grade 5 and not only, there are natural, integer, simple and compound numbers.
You can consider the following sets:
- integers;
- letters of the alphabet;
- primary coefficients;
- triangles with different values of the sides.
It can be seen that these examples are clearly defined sets of objects. Let's look at a few more examples:
- five most famous scientists of the world;
- seven beautiful girls in society;
- three best surgeons.
These examples of the power of a multitude are not clearly defined collections of objects, because the criterion of the “most famous,” “most beautiful,” “best” varies from person to person.
Sets
This value represents a clearly defined number of different objects. Assuming that:
- a set of words is a synonym, aggregate, class and contains elements;
- objects, members are equal in meaning terms;
- sets are usually denoted in capital letters A, B, C ;
- elements of the set are represented in small letters a, b, c.
If “a” is an element of the set A, then it is said that “a” belongs to A. Denote the phrase “belongs” by the Greek symbol “∈” (epsilon). Thus, it turns out that a ∈ A. If 'b' is an element that does not belong to A, it is represented as b ∉ A. Some important sets used in class 5 mathematics are represented using the following three methods:
- applications;
- registries or tabular;
- construction creation rule.
Upon closer examination, the application form is based on the following. In this case, a clear description of the elements of the set is given. All of them are enclosed in braces. For instance:
- the set of odd numbers less than 7 is written as {less than 7};
- a set of numbers greater than 30 and less than 55;
- the number of class students whose weight is greater than teachers.
In the registry (tabular) form, the elements of a set are listed in a pair of brackets {} and separated by commas. For instance:
- Let N denote the set of the first five natural numbers. Therefore, N = → the registry form
- Set of all vowels of the English alphabet. Therefore, V = {a, e, i, o, u, y} → the registry form
- The set of all odd numbers is less than 9. Therefore, X = {1, 3, 5, 7} → the registry form
- The set of all letters in the word "Mathematics". Therefore, Z = {M, A, T, H, E, I, C, S} → The registry form
- W is the set of the last four months of the year. Therefore, W = {September, October, November, December} → the registry.
It is worth noting that the order in which the elements are listed does not matter, but they should not be repeated. The established form of construction, in the given case, the rule, the 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 one property in order to become a member of the value in question.
In this form of representing a set, an element of the set is described using the symbol “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 form of set-builder is written as - {countable number and greater than 12}. It will be read in a certain way. That is, "P is the set of elements of x, such that x is a countable number and greater than 12".
Solved example using three methods for representing a set: the number of integers between -2 and 3. The following are examples of different types of sets:
- An empty or null set that does not contain any element and is denoted by and is read as phi. In the form of a list, ∅ has the spelling {}. A finite set is empty, since the number of elements is 0. For example, a set of integer values is less than 0.
- Obviously, they should not be <0. Therefore, it is an empty set.
- A collection containing only one variable is called a singleton set. It 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 a rainbow.
An infinite amount is a set. Items in it cannot be listed. That is, containing such variables is called an infinite set. Examples:
- the power of the set of all points in the plane;
- set of all prime numbers.
But it is worthwhile to understand that all the powers of combining sets cannot be expressed in the form of a list. For example, real numbers, since their elements do not correspond to any particular scheme.
The cardinal number of a set is the number of different elements in a given quantity A. It is denoted by n (A).
For instance:
- 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 comparing sets
Two powers of the 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 the set A and B if they contain the same elements. Each coefficient from A is a variable from B, and each from B is a specified value of A. Therefore, A = B. Various types of union of sets in cardinality and their definitions are explained using the indicated examples.
The essence of finiteness and infinity
What are the differences between the power of a finite set and an infinite?
The first name is characterized by the following name if it is either empty or has a finite number of elements. In a finite set, a variable can be indicated if it has a limited account. For example, using a natural number 1, 2, 3. And the listing process ends on some N. The number of different elements counted in a finite set S is denoted by n (S). It is also called order or cardinal. Symbolically indicated by 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 listed. If it has an unbounded (i.e. uncountable) natural number 1, 2, 3, 4 for any n. A set that is not finite is called infinite. Now we can discuss examples of the considered numerical values. Final 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 whose modulus is 9}. Then S = {-9, 9} is a finite set and n (S) = 2.
- Set of all people.
- The number of all birds.
Examples of infinite set:
- the number of existing points on the plane;
- the number of all points in the line segment;
- the set of positive integers that are multiples of 3 is infinite;
- all integers and natural numbers.
Thus, from the above reasoning it is clear how to distinguish between finite and infinite sets.
Power set continuum
If we compare the set and other existing values, then the complement is attached to the set. If ξ is universal and A is a subset of ξ, then the complement to A is the number of all elements ξ that are not elements of A. The complement A with respect to ξ is symbolically denoted by 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 a power continuum has the following features:
- the complement of the universal quantity is the empty value in question;
- this zero-set variable is universal;
- the quantity and its complement are disjoint.
For instance:
- Let the number of natural numbers be a universal set and A be even. Then, then A '{x: x is an odd set with the same digits}.
- Let ξ = the set of letters in the alphabet. A = consonant set. Then A '= number of vowels.
- An addition to the universal set is an empty quantity. It can be denoted by ξ. Then ξ '= The set of those elements that are not included in ξ. An empty set φ is written and denoted. Therefore, ξ = φ. Thus, the complement to the universal set is empty.
In mathematics, the “continuum” is sometimes used to denote a real line. And more generally, to describe such objects:
- continuum (in set theory) - real line or corresponding cardinal number;
- linear - any ordered set that shares certain properties of a real line;
- the continuum (in topology) is a nonempty 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), theory or model that explains the gradual transitions from one state to another without any drastic changes.
Unification and intersection problems
It is known that the intersection of two or more sets is a quantity containing all the elements that are common in these values. Word tasks on sets are solved to get the basic ideas on how to use the properties of union and intersection of sets. The solved basic problems of words on sets 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, there is n (A ∩ B).
Connection 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 a shaded area representing A ∩ B.
- The difference of the two sets can be represented by Venn diagrams. With a shaded area representing A - B.
- The relationship between the three sets using the Venn diagram. If ξ represents a universal quantity, then A, B, C are three subsets. Here, all three sets are overlapping.
Compilation of information about the set
The power 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, the power of many examples are the following:
Let A = {0,1,2,3} | | = 4, where | A | represents the power of A.
Now you can find your set of power. It is also quite simple. As already mentioned, the power set is installed from all subsets of a given number. Therefore, you need to basically define all the variables, elements and other values of A that 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 the power finds 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 power of the set A = 16. Obviously, this is a tedious and cumbersome method of 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 quantity. | P | = 2 ^ N, where N is the number of elements in some A. This formula can be obtained using simple combinatorics. Thus, the question is 2 ^ 11, since the number of elements in the set A is 11.
So, the set is any numerically expressed amount, which can be all kinds of objects. For example, cars, people, numbers. In mathematical terms, this concept is wider and more generalized. If at the initial stages the numbers and options for solving them are sorted out, then in the middle and higher stages the conditions and tasks are complicated. In fact, the power of combining a set is determined by the belonging of an object to any group. That is, one element belongs to the class, but has one or more variables.