Abstract concepts and symbolic terms are the salient features of set units, such as the concepts of intersection, union and complement and their representations, the relationship between sets and elements and their representations, the relationship between sets and representations, the definitions of subsets, proper subset and sets, and so on. These concepts, relationships and representations can be used as the basis, starting point and even breakthrough point to solve the set problem. Therefore, to learn the content of set well, we must accurately grasp the concept of set and skillfully use the relationship between set and set to solve specific problems.
Second, pay attention to understand the nature of the elements of the set, and learn to use the element analysis method to investigate the related problems of the set.
As we all know, a set can be regarded as the sum of some objects, and each object is called an element of this set. The elements in the collection have "three attributes":
(1), certainty: the elements in the set should be certain and cannot be ambiguous.
(2) Reciprocity: The elements in the set should be different from each other, and the same element can only be counted as one in the set.
(3) Disorder: The elements in the set are out of order.
Third, understand the mathematical thinking method contained in the set problem and master the basic law of solving the set problem.
Fourth, pay attention to the particularity of empty set, so as to prevent the problem-solving mistakes caused by ignoring the special situation of empty set.
Things that are certain and distinguishable within a certain range, as a whole, are called sets, or elements for short. Any set is a subset of itself.
Relationship between elements and collections:
There are two relationships between elements and sets: attribution and non-attribution.
Set classification:
Union set: The set whose elements belong to A or B is called the union (set) of A and B, marked as A∪B (or B∪A), and pronounced as A and B (or B and A), that is, A∪B={x|x∈A, or X.
Intersection: The set with elements belonging to A and B is called the intersection (set) of A and B, marked as A∩B (or B∩A), and read as "A crosses B" (or "B crosses A"), that is, A∩B={x|x∈A, X ∩.
For example, the complete works U = {1, 2,3,4,5} A = {1,3,5} B = {1,2,5}. Then because both A and B have 1, 5, A ∩ B = {1, 5}. Let's take another look. Both contain 1, 2, 3, 5, no matter how much, either you have it or I have it. Then say a ∪ b = {1, 2, 3, 5}. The shaded part in the figure is a ∩ B.
Infinite set: Definition: A set containing infinite elements in a set is called an infinite set.
Finite set: let N+ be a positive integer, Nn={ 1, 2,3, ..., n}. If there is a positive integer n that makes the set A correspond to NN one by one, then A is called a finite set.
Difference: The set of elements belonging to A but not to B is called the difference between A and B (set).
Note: An empty set is contained in any set, but it cannot be said that an empty set belongs to any set.
Complement set: A set consisting of elements belonging to the complete set U but not to the set A is called the complement set of the set A, and is denoted as CuA, that is, CuA={x|x∈U, and x does not belong to A}
An empty set is also considered a finite set.
For example, if the complete sets U = {1, 2, 3, 4, 5} and A = {1, 2, 5}, then 3,4 in the complete set but not in A is CuA, which is the complement of A. CUA = {3,4}.
In information technology, CuA is often written as ~ a.
When some specified objects are gathered together, they become a set, which contains finite elements and infinite elements. An empty set is a set without any elements, and it is recorded as φ. An empty set is a subset of any set and a proper subset of any non-empty set. Any set is a subset of itself, and both subset and proper subset are transitive.
Explanation: If all elements of set A are elements of set B at the same time, call A a subset of B, and write A? B. If A is a subset of B and A is not equal to B, call A proper subset of B and write A? B.
Representation methods of sets: enumeration and description are commonly used.
1. enumeration: usually used to represent a finite set. All the elements in the collection are listed one by one and enclosed in braces. This method of representing a set is called enumeration. { 1,2,3,……}
2. Description: It is often used to represent an infinite set. The public * * * attribute of the elements in the collection is described by words, symbols or expressions and enclosed in braces. This method of representing a set is called description. {x|P}(x is the general form of the elements of this set, and p is the * * * same property of the elements of this set) For example, a set composed of positive real numbers less than π is expressed as {x | 0.
3. Schema Method (venn diagram): In order to visually represent a set, we often draw a closed curve (or circle) and use its interior to represent a set.
4. Natural language
Symbols of commonly used number sets:
(1) The set of all non-negative integers is usually called the set of non-negative integers (or the set of natural numbers), and is recorded as n.
(2) Exclude the set of 0 from the set of non-negative integers, also known as the set of positive integers, and record it as N+ (or N*).
(3) The set of all integers is usually called the set of integers, and is denoted as z..
(4) The set of all rational numbers is usually referred to as the rational number set for short, and is recorded as Q..
(5) The set of all real numbers is usually called the set of real numbers, and is denoted as r.
(6) the complex set is c.