Analysis:
Related concepts of * * *:
It is composed of some numbers, some points, some figures, some algebraic expressions, some objects and some people. We say that all the objects in each group form a * * *, or that some specified objects together become a * * *, also known as a set. Every object in * * * is called an element of this * * *.
Definition: Generally speaking, some specified objects together become a * * *.
The concepts of 1 and * * *
(1) * *: some specified objects are grouped together to form a * * * (set for short).
(2) Element: Every object in * * * is called this * * * element.
2, commonly used digital sets and symbols
(1) non-negative integer set (natural number set): all non-negative integers are marked with n,
(2) Positive integer set: the set without 0 in the non-negative integer set is recorded as N* or N+
(3) Integer set: all integers are marked as z,
(4) Rational number set: all rational numbers are marked with q,
(5) Real number set: the * * * of all real numbers is recorded as R.
Note: (1) natural number set is the same as non-negative integer set, that is, natural number set includes.
Count 0
(2) A set that does not contain 0 in a non-negative integer set is denoted as N* or N+ Q, z, r, etc.
A set that excludes 0 from a number set is also expressed in this way, such as excluding 0 from an integer set.
Group, denoted as Z*
3. Elements and * * *
(1) belongs to: If A is an element of *** A, it is said that A belongs to A and is marked as A ∈ A
(2) Does not belong to: If A is not an element of *** A, it is said that A does not belong to A, and it is recorded as
4. Characteristics of elements in * *
(1) determinism: given an element or according to a clear standard in this * * *,
Or not, not ambiguous.
(2) Reciprocity: the elements in * * * are not repeated.
(3) Disorder: The elements in * * * have no certain order (usually written in normal order).
5. (1) * * is usually represented by capitalized Latin letters, such as A, B, C, P and Q. ...
Elements are usually represented by lowercase Latin letters, such as A, B, C, P, Q. ...
(2) The opening direction of "∈" should not be written as A ∈ A in reverse.
In addition:
Core knowledge
1.***
Concepts such as point, line and surface are primitive and undefined in geometry, while * * * is primitive and undefined in * * * theory. Usually, some specified objects together become a * * *, also known as a collection. Braces are generally used to indicate * *, such as "cars, planes, ships" and other means of transportation.
2. Elements in * *
Every object in * * * is called this * * * element. For example, the elements of "Chinese municipality" are: Beijing, Shanghai, Tianjin and Chongqing.
The elements in * * * usually use lowercase Latin letters a, b, c, …
If A is an element of *** A, it is said that A belongs to *** A, and it is marked as A ∈ A;
If A is not an element of *** A, it is said that A does not belong to *** A and is recorded as A. 。
3. Characteristics of elements in * *
(1) certainty For *** A and an object X, there is a clear standard to judge whether it is x∈A or x A, and they must be one of them without ambiguity.
Things like "famous mathematicians" and "beautiful people" generally cannot constitute * * * in the mathematical sense, because there is no clear standard to judge whether each specific object belongs to * * *.
(2) reciprocity. For a given * * *, any two elements are different; Therefore, the same element in * * * can only be counted as one. For example, two isoroots of the equation x2-2x+ 1=0, x 1 = x2 = 1, are marked by * * as {1} instead of {1.
(3) The elements in unordered * * * are out of order, for example, * * {1,2} and {2, 1} are the same * * *, but they are actually written in a certain order, for example, {- 1, 65433}.
4.*** symbol
(1) Enumerates all the elements in * * * and writes them in braces.
(2) Description method uses description to express * * *, and it is necessary to accurately understand the attributes of its elements. For example, * * {y | y = x2} represents all values of the function y, that is, {y | y≥0 }；; * * * {x | y = x2} indicates all values of the independent variable x, that is, {x | x is an arbitrary real number}; * * * {x, y | y=x2} represents all points on the parabola y=x2 and is a point set (parabola); And * * * {y=x2} is a single-element set expressed by enumeration, that is, a finite set with only one element (equation y=x2).
(3) Graphical method In order to express * * * intuitively, we often draw a closed curve and use its interior to represent a * * *. For example, as shown in the figure, it can be expressed as * * {1, 2,3,4}.
5. Specific * * * symbols
A set of natural numbers (or a set of non-negative integers), marked as n, and a set excluding 0 from the set of natural numbers, also known as a set of positive integers, marked as N* or N+ (note that the set of natural numbers includes 0);
Integer set, denoted as z;
Set of rational numbers, recorded as q;
Set of real numbers, denoted as r;
The sets without 0 in the sets of Z, Q, R, etc. are denoted as Z* (or Z+), Q* (or Q+) and R* (or R+) respectively.
6. Classification of * *
① Finite set: * * A finite element is called a finite set. For example, a = {1, 2,3,4}
② Infinite set: * * An infinite set contains infinite elements. For example, *** N+
③ Empty set: * * A set without any elements is called an empty set. For example, the solution set of equation x2+2x+3=0 in the real number range.