The meaning of 1. set
2. Three characteristics of elements in a set:
(1) elements, such as mountains in the world.
(2) The mutual anisotropy of elements, such as the set of happy letters {H, a, p, Y}.
(3) The disorder of elements: for example, {a, b, c} and {a, c, b} represent the same set.
3. Representation of assembly: {…} For example, {basketball players in our school}, {Pacific Ocean, Atlantic Ocean, Indian Ocean, Arctic Ocean}
(1) The set is expressed in Latin letters: A={ basketball players in our school}, B={ 1, 2, 3, 4, 5}.
(2) Representation methods of sets: enumeration method and description method.
Note: Commonly used digit 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
1) enumeration: {A, b, C...}
2) Description: describes the common attributes of the elements in the set, and is written in braces to indicate the set {x? r | x-3 & gt； 2}，{ x | x-3 & gt； 2}
3) Language description: Example: {A triangle that is not a right triangle}
4) Venn diagram:
4, the classification of the set:
The (1) 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}
Second, the basic relationship between sets
1. "Inclusive" relation-subset
Note: There are two possibilities.
(1)A is a part of B;
(2)A and B are the same set.
On the other hand, set A is not included in set B, or set B does not include set A, which is marked as AB or BA.
2. "Equality" relation: A=B(5≥5, and 5≤5, then 5=5) holds.
Example: let a = {x | x2-1= 0} b = {-1,1} "Two sets are equal if their elements are the same".
Namely:
(1) Any set is a subset of itself. Aiya
② proper subset: If AíB and A 1B, then set A is the proper subset of set B, and it is recorded as AB (or BA).
③ If aí b and bí c, then aí c.
④ If AíB and BíA exist at the same time, then a = b.
3. A set without any elements is called an empty set and recorded as φ.
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.
4. Number of subsets:
A set of n elements, including 2n subsets, 2n- 1 proper subset, 2n- 1 nonempty subset and 2n- 1 nonempty proper subset.
Third, the operation of the set.
Intersection complement set of operation types
Define a set consisting of all elements belonging to A and B, which is called the intersection of A and B, and it is marked as AB (pronounced as' A crosses B'), that is, AB={x|xA, and XB}.
A set consisting of all elements belonging to set A or set B is called the union of A and B, and it is marked as AB (pronounced as' A and B'), that is, AB={x|xA, or xB}).
Special knowledge points of mathematics II. Summary of knowledge points
The related concepts of 1, set.
1) Set: set some specified objects together to form a set. Each object is called an element.
note:
① Set and its elements are two different concepts, which are given by description in textbooks, similar to the concepts of points and lines in plane geometry.
② The elements in a set are deterministic, different and disorderly ({a, b} and {b, a} represent the same set).
③ A set has two meanings: all eligible objects are its elements; As long as it is an element, you must sign the condition.
2) Representation methods of sets: enumeration method, description method and graphic method are commonly used.
3) Classification of sets: finite set, infinite set and empty set.
4) Common number set: n, z, q, r, N*
2. Concepts such as subset, intersection, union, complement, empty set and complete set.
1) subset: if there is x∈B for x∈A, then A B (or ab);
2) proper subset: A B has x0∈B but x0 A；; Marked as b (or, and)
3) intersection: A∩B={x| x∈A and x∈B}
4) and: A∪B={x| x∈A or x∈B}
5) Complement set: cua = {x | x but x ∈ u}
3. Understand the relationship between sets and elements, sets and sets, and master relevant terms and symbols.
4. Several equivalence relations about subsets.
①A∩B = A A B； ②A∪B = B A B； ③A B C uA C uB；
④A∩CuB = empty set cuab; ⑤CuA∪B=I A B .
5. Attributes of intersection and union operations
①A∩A=A，A∩B = B∩A； ②A∪A=A，A∪B = B∪A；
③Cu(A∪B)= CuA∪CuB，Cu(A∪B)= CuA∪CuB；
6. Number of finite subsets: If the number of elements in set A is n, then A has 2n subsets, 2n- 1 nonempty subset and 2n-2 nonempty proper subset.
Second, the integration of knowledge points
The sum total of things with certain properties. The "thing" here can be a person, an object or a mathematical element. For example:
1, scattered people or things get together; Assemble: urgent.
2. Mathematical terminology. A group of mathematical elements with a certain isomorphism: rational numbers.
3. Slogans and so on. Set has many concepts in mathematical concepts, such as set theory: set is the basic concept of modern mathematics, and the theory that specializes in set is called set theory. Cantor (G.F.P, 1845- 19 18), a pioneer of German mathematicians, is the founder of set theory. At present, the basic idea of set theory has penetrated into all fields of modern mathematics.
Set is a basic concept in mathematics. What are the basic concepts? Basic concepts are concepts that cannot be defined by other concepts. The concept of set can be defined in an intuitive and axiomatic way.
Collection is to bring together some definite and distinguishable objects in people's intuition or thinking to make them a whole (or monomer). This whole thing is a set. Those objects that make up a set are called elements of this set (or simply elements).
The relationship between elements and sets
There are two relationships between elements and sets: attribution and non-attribution.
Third, the relationship between set and set
When some specified objects are gathered together, they become a set of symbols. The finite elements contained in the set are called finite sets, the infinite elements are called infinite sets, and the empty set is a set without any elements, which 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. Subsets 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, then A is called proper subset of B, which is generally written as A? B. What will be in the middle school textbook? A ≠ symbol (as shown on the right) is added below the symbol, so don't confuse it. The exam should be based on the textbook. A collection of all people is a collection of all people, proper subset. 』
Several algorithms of sets
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.
A set with elements is called the intersection of A and B, marked as A∩B (or B∩A) and pronounced as "A ∩ B" (or B ∩ A "), that is, A∩B={x|x∈A, X. 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 picture is a ∩ B. Interestingly; For example, how many numbers in 1 to 105 are not integer multiples of 3, 5 and 7? The result is that the subtraction set of each item of 3, 5 and 7 is 1 and then multiplied. 48. Symmetric difference set: Let A and B be sets, and the symmetric difference set A of A and B? The definition of b is: a? B =(A-B)∩(B-A) For example: A={a, b, c}, B={b, d}, then A? Another definition of B={a, c, d} symmetric difference operation is: a? B =(A∪B)-(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, N_n={ 1, 2, 3, ..., n}, if there is a positive integer n, the difference is that the elements belong to. Note: An empty set is contained in any set, but it cannot be said that "an empty set belongs to any set". Complement set is a concept derived from difference set, which means that a set composed of elements belonging to complete set U but not to set A is called the complement set of set A, and it is denoted as CuA, that is, an empty set with CuA={x|x∈U and x not belonging to A} is also considered as 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,
Fourthly, the nature of set elements.
1. Certainty: Every object can determine whether it is an element of a set. Without certainty, it cannot be a set. For example, "tall classmates" and "small numbers" cannot form a set. This property is mainly used to judge whether a set can constitute a set.
2. Independence: The number of elements in the set and the number of the set itself must be natural numbers.
3. Relevance: Any two elements in the set are different objects. If written as {1, 1, 2}, it is equivalent to {1, 2}. Being different from each other makes the elements in the collection not repeat. When two identical objects are in the same set, they can only be counted as an element of this set.
4. Disorder: {a, B, c}{c, B, a} are the same set.
The collection has the following properties
If a is included in b, then A∩B=A and a ∪ b = b
Representation method of set
Sets are usually represented by uppercase Latin letters, such as: a, b, c… while elements in sets are represented by lowercase Latin letters, such as: a, b, C… Latin letters are just equivalent to the names of sets and have no practical significance. The method of assigning Latin letters to a set is represented by an equation, for example, in the form of A={…}. The left side of the equal sign is capitalized Latin letters, and the right side is enclosed in curly braces. In brackets are some mathematical elements with the same nature.