What is a set of real numbers and what is a set of real numbers?

In layman's terms, a set of all rational and irrational numbers is a set of real numbers, usually represented by a capital letter r. In the 18th century, calculus was developed on the basis of real numbers. But there was no precise definition of the set of real numbers at that time.

It was not until 1871 that the German mathematician Cantor first proposed a strict definition of real numbers. The definition is based on four sets of axioms: paragraph 1, axioms of addition:

For any element a and b that belong to the set r, you can define their addition a+b, and a+b belongs to r; The addition has a constant element 0, and a+0=0+a=a (so there is an opposite number); Addition has a commutative law, a+b=b+a; Addition has an associative law, (a+b)+c=a+(b+c). 2. Axiom of multiplication: For any element a and b that belongs to the set r, it is possible to define their multiplication a·b, and a·b belongs to r; The multiplication has a constant element 1, and a·1=1·a=a (so that there is a reciprocal in addition to 0); Multiplication has a commutative property, a·b=b·a; Multiplication has a associative property, (a·b)·c=a· (b·c)； Multiplication has a distribution rate for addition, i.e., a· (b+c)=(b+c)·a=a·b+a·c。

3. The axiom of order: any x and y belong to r, and only one of xy is true; If x0, then x·z 4, the complete axioms: (1) Any set of non-empty upper bounds (contained in r) must have upper bounds. (2) Let a and b be two sets contained in r, and for any x that belongs to a and y belongs to b, any set of x that satisfies the above four sets of axioms is called a set of real numbers, and the elements of the set of real numbers are called real numbers.

1. Addition axioms: for any element a and b that belong to the set r, it is possible to define their addition a+b, and a+b belongs to r; The addition has a constant element 0, and a+0=0+a=a (so there is an opposite number); Addition has a commutative law, a+b=b+a; Addition has an associative law, (a+b)+c=a+(b+c). 2. Axiom of multiplication:

For any element a and b that belong to the set r, it is possible to define their multiplication a·b, and a·b belongs to r; The multiplication has a constant element 1, and a·1=1·a=a (so that there is a reciprocal in addition to 0); Multiplication has a commutative property, a·b=b·a; Multiplication has a associative property, (a·b)·c=a· (b·c)； Multiplication has a distribution rate for addition, i.e., a· (b+c)=(b+c)·a=a·b+a·c。 3. The axiom of order: any x and y belong to r, and only one of xy is true; If x0, then x·z and if x (2) let a and b be two sets contained in r, and for any x they belong to a and y belong to b, there are x< y, then there must be c to r, such that for any x that belongs to a and y belongs to b, there is x

Look at the textbook, high school math must be one.

The set of real numbers greater than or equal to -3 and less than 11 is.

Sets, abbreviated as sets, are a basic concept in mathematics and are also set simplifications.

The main object of study. The basic theory of set theory was created in the 19th century.

The simplest way to say about a set is the definition in naïve set theory (the most primitive set theory), that is, a set is "a definite set of things", and the "things" in a set are called elements. A modern set is generally defined as a whole made up of one or more definite elements.

Integer: An integer is a collective term for all numbers in a sequence, including negative integers, zeros (0), and positive integers.

and natural numbers. Similarly, an integer is an infinite set of countables.

This set is mathematically usually represented as a bold Z or, derived from the initial letter of the German word zahlen, which means "number".

In algebraic number theory, these general integers that are rational numbers are called rational integers to distinguish them from concepts such as Gaussian integers.

"All real numbers", "set of real numbers", which are themselves a representation of the set, can no longer be parenthesed.

If it is written, then it is not a set of real numbers, but a set of sets.

is a set of elements, and the elements of this set are the set of real numbers.

If an equivalence relation is given on the set x, then the set of all equivalence classes forms a partition of x. Conversely, if a partition p is given on x, it is possible to define an equivalence relation written as x y on x if and only if there is a member of p that contains both x and y. The concepts of "equivalence" and "division" are therefore essentially equivalent.

In this way, the words "all real numbers", "set of real numbers", which are themselves a representation of the set, cannot be bracketed.

If it is written, then it is not a set of real numbers, but a set of sets.

is a set of elements, and the elements of this set are the set of real numbers.

Of course, the set of real numbers can also be represented in this way, or in this case, parentheses should be enlarged.

For example, a set of positive numbers can be written as "set of positive numbers" or "all positive numbers" in words, without parentheses.

It can also be expressed as the need to enlarge parentheses at this point.

A set is the convergence of certain distinguishable objects in people's intuition or thinking to make it a whole (or monomer), and this whole is a set. Those objects that make up a set are called elements (or simply metas) of the set.

So it doesn't specify what the elements in the set are, anything can be included in the set, and the elements in it can even be students, basketballs, or whatever. However, if the emphasis is on the set of numbers, the elements in it can of course only be numbers, but they can also include imaginary numbers. For example, we usually use r to represent the set of real numbers, and c to represent the set of complex numbers (complex numbers are the sum of real and imaginary numbers), then of course there are imaginary numbers in c.

1. Addition axioms: for any element a and b that belong to the set r, it is possible to define their addition a+b, and a+b belongs to r;

The addition has a constant element 0, and a+0=0+a=a (so there is an opposite number);

Addition has a commutative law, a+b=b+a;

Addition has an associative law, (a+b)+c=a+(b+c). 2. Axiom of multiplication: For any element a and b that belongs to the set r, it is possible to define their multiplication a·b, and a·b belongs to r;

The multiplication has a constant element 1, and a·1=1·a=a (so that there is a reciprocal in addition to 0);

Multiplication has a commutative property, a·b=b·a;

Multiplication has a associative property, (a·b)·c=a· (b·c)；

Multiplication has a distribution rate for addition, i.e., a· (b+c)=(b+c)·a=a·b+a·c。 3. The axiom of order: any x and y belong to r, and only one of xy is true;

If x0, then x·z transitivity: if x(2) lets a and b be two sets contained in r, and for any x that belongs to a and y belongs to b, any set of x that satisfies the above four sets of axioms is called a set of real numbers, and the elements of the set of real numbers are called real numbers.

Strictly written, it should be written as a real number.

But we usually say it in the field of real numbers, because the context is too obvious to cause misunderstanding, and can be omitted.

And the last one, normally we would write, because there is a complex solution to x, and if you don't write it, there may be ambiguity. Of course, the worst thing is to get mixed up and accidentally quote the intermediate results to produce ambiguity, resulting in an unreliable reasoning process).

Mathematics is also a language, and it can also be abbreviated and abbreviated, as long as it does not cause ambiguity. Most of the places where it appears may involve complex numbers, otherwise, if it must be in the field of real numbers, we will simply abbreviate it as an empty set

It doesn't matter, depending on your habits, I usually write x r.

If it is known that the slag is empty and the slag is a=, b=, if a b = is empty, find the value range of the bucket beam trace of the empty and real number k.

k+1>2k-1

K2K+1>5 or 2K-14 or K4 In summary. The value range of the real number k is k4

A number whose logarithm is a known number is said to be the true number of a known number. True numbers, also known as antinumbers, are numbers that are relative to false numbers (i.e., logarithms). It was first seen in the "Essence of Mathematics" under the volume 38 "Logarithmic Proportion".

Let a be a positive number that is not equal to 1, i.e., a>0, and a ≠ 1. If ap=b, then p is said to be the logarithm of b with base a; And b is called p, and the true number with a as the base is called. Denote as p=logab.

For example, with 2 as the base, the logarithm of 8 is 3 and the true number of 3 is 8

Resources.

Hello! There are three elements of the set element: certainty, mutuality, and disorder.

Two pairs are not equal, a total of three cases:

1≠ x, x ≠ x 2-x solution gives x≠0 and x≠2

1≠ x 2-x gives x≠(1+ 5) 2, and x≠(1- 5) 2

In summary, the set of x values is.

You're missing the third case.

If the set a= , then the set of real numbers x is

1≠x ..x≠1

1≠x²-x...x≠(1+√5)/2,x≠(1-√5)/2x≠x²-x...x≠0,x≠2

x≠0,x≠1,x≠2,x≠(1+√5)/2,x≠(1-√5)/2

There are 3 inequalities in total.

x≠1 1x^2-x≠x 2

x^2-x≠1 3

There are 2 solutions for x≠0 x≠2

There are 3 solutions of x≠(1+root5)2 and x≠(1-root5)2