I. Collection of related concepts
1, meaning of set: some specified objects are set together into a set, and each object is called an element.
2. Three characteristics of elements in a set:
The certainty of (1) elements; ② Mutual anisotropy of elements; ③. Disorder of elements
Description: (1) For a given set, the elements in the set are certain, and any object is either an element of the given set or not.
(2) In any given set, any two elements are different objects. When the same object is contained in a collection, it is only an element.
(3) The elements in the set are equal and have no order. So to judge whether two sets are the same, we only need to compare whether their elements are the same, and we don't need to examine whether the arrangement order is the same.
(4) The three characteristics of set elements make the set itself deterministic and holistic.
3, the classification of the set:
1. The finite set contains a set of finite elements.
2. An infinite set contains an infinite set of elements.
3. An example of an empty set without any elements: {x | x2 =-5}
4. Representation of assembly: {} For example, {basketball players in our school}, {Pacific Ocean, Indian Ocean, Arctic Ocean}
1.Set is expressed in Latin letters: A={ basketball player of our school }B={ 12345}
2. Representation methods of sets: enumeration and description.
Note: Commonly used number sets and their symbols:
The set of nonnegative integers (i.e. natural number set) is recorded as n.
Positive integer set N* or N+ integer set z rational number set q real number set r
On the concept of attribution
Elements in a collection are usually represented by lowercase Latin letters. For example, if A is an element of set A, it means that A belongs to set A, and it is denoted as AA; On the other hand, if a does not belong to the set a, it is recorded as a? A
Enumeration: enumerate the elements in the collection one by one, and then enclose them in braces.
Description: A method of describing the common attributes of elements in a collection and writing them in braces to represent the collection. Use certain conditions to indicate whether some objects belong to the collection.
① Language Description: Example: {A triangle that is not a right triangle}
② Description of mathematical expression: Example: The solution set of inequality x-32 is {x? R| x-32} or {x| x-32}
Second, the basic relationship between sets
1. contains a subset of relationships.
Note: There are two possibilities that A is a part of B (1); (2)A and B are the same set.
On the other hand, set A is not contained in set B or set B does not contain set A, and set A is marked as A B or B A.
2. A set without any elements is called an empty set, and it is recorded as
It is stipulated that an empty set is a subset of any set and an empty set is a proper subset of any non-empty set.
3. Equation relation (55 and 55, then 5=5)
Example: let a = {x | x2-1= 0} b = {-11} elements be the same.
Conclusion: For two sets A and B, if any element of set A is an element of set B and any element of set B is an element of set A, we say that set A is equal to set B, that is, A = B.
(1) Any set is a subset of itself. Answer? A
② proper subset: If a? B and a? B then says that set A is the proper subset of set B, and writes it as A B (or B A).
3 if a? B B？ C so a? C
4 if a? At the same time? Then A=B
Third, the operation of the set.
Definition of 1. union: Generally speaking, a set consisting of all elements belonging to set A or set B is called the union of AB. Note: AB (pronounced as A and B) means AB={x|xA, or xB}.
2. Definition of intersection: Generally speaking, the set of all elements belonging to A and B is called the intersection of AB.
Write AB (pronounced a to b), that is, AB={x|xA, and xB}.
3. Complete works and supplements
(1) Complement set: Let S be a set and A be a subset of S (that is, a set composed of all elements in S that do not belong to A), which is called the complement set (or complement set) of subset A in S..
Note: CSA is CSA ={x? x？ S and x? A}
(2) Complete Works: If the set S contains all the elements of each set we want to study, this set can be regarded as a complete set. Usually represented by u.
(3) Properties: ⑴ cu (cua) = a ⑴ (cua) ⑴ (cua) a = u.
4. the nature of intersection and union: AA = A A= B = BA, aa = a.
A= A AB = BA。