Things that are certain and distinguishable within a certain range, as a whole, are called sets, or elements for short. Such as (1) different Chinese characters appearing in the true story of Ah Q (2) all English capital letters. 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 ∩.
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.
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? 6? 7 B. If A is a subset of B and A is not equal to B, call A the proper subset of B and write A? 6? 3 B .
A collection of all people is a collection of all people, proper subset. 』
The nature of the set:
Determinism: Every object can determine whether it is an element of a set. Without certainty, there will be no trap. For example, "tall classmates" and "small numbers" cannot form a set.
Correlation: Any two elements in a collection are different objects. It cannot be written as {1, 1, 2}, but as {1, 2}.
Disorder: {a, b, c}{c, b, a} are the same set.
A set has the following properties: if A is contained in B, A∩B=A, A ∪ B = B
Representation methods of sets: enumeration and description are commonly used.
1. Enumeration: commonly used to represent a finite set, in which all elements are listed one by one and written 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: In order to visually represent a set, we often draw a closed curve (or circle) and use its interior to represent a set.
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.
Operation of sets:
1. commutative law
A∩B=B∩A
A∪B=B∪A
2. Association law
(A∩B)∩C=A∩(B∩C)
(A∪B)∪C=A∪(B∪C)
3. Distribution law
A∩(B∪C)=(A∪B)∩(A∪C)
A ∪( B∪C)=(A∪B)∪( A∪C)
2 De Morgan's Law
Cs(A∩B)=CsA∪CsB
cs(A∪B)= CsA∪CsB
3 "Exclusion principle"
When we study a set, we will encounter problems about the number of elements in the set. We write the number of elements in finite set A as card(A). For example, A={a, b, c}, then card (A)=3.
Card (A∪B)= Card (A)+ Card (B)- Card (A∪B)
Card (A∪B∪C)= Card (A)+ Card (B)+ Card (C)- Card (A∪B)- Card (C∪A)+ Card (A ∪.
1985 Cantor, a German mathematician and founder of set theory, talked about the word set. Enumeration and description are common methods to represent collections.
Law of absorption
A ∨( A∩B)= A
A∩(A∪B)=A
Supplementary law
A∪CsA=S
a∩CsA =φ
[answer]
Understand the concept of set, the nature of set, the representation of elements and sets and their relationships.
Significance and application of son, intersection, union and complement of set. Master related terms and symbols, and accurately use set language to express, study and deal with related mathematical problems.
[difficulties]
The meanings of various concepts about set and the differences and connections between these concepts.
Accurately understand and apply more new concepts and symbols to deal with mathematical problems.