Note: This article** is an article created by Toutiao himself today, because Toutiao is a platform that pays attention to news gossip, and it is not very supportive of mathematical creation, I am in order to have a systematic work, so I turned it around, and I finally selected Baijia to create.
The basic relationship between sets
1. Definition and representation of subsets, true subsets, and sets equally
Subset:If any element in the set is an element of the set, then the set is called a subset of the set, denoted as or, read as "contained" (or "contained").
Note:
When a collection is not included in a collection, it is denoted as or.
The subset can be represented by the following graph:
Venn diagram representation of a subsetDescription: The collection itself can be a subset of itself, ie
True Subset:If a collection is a subset of a collection, and at least one element in it does not belong to, then the collection is called a true subset of the collection. Write as: or.
The true subset diagram is represented as follows:
True Subset Venn Diagram Through the introduction of subsets and subsets, students should pay attention to the following points:
A subset is a depiction of the relationship between two sets, and it reflects the relationship between the part and the whole (while the relationship between the elements and the set is the relationship between the individual and the whole).
Not all two collections have an inclusion relationship
For example: =, =, because, but, so is not a subset; In the same way, because, but the element, so is not a subset
The relationship between analogy and is like the relationship between (less than or equal to) and (less than), which is less than or equal to, which means less than.
For example: what is right is wrong, and if it is, it is also true; In contrast: right, but wrong; If so, then it is also true.
The sets are equal
If it is a subset of a collection, and the collection is a subset of the collection, the collection is equal to the collection
That is. 2. Empty set
Definitions:In general, we call a set that does not contain any elements an empty set and denote it as.
and stipulates that an empty set is a subset of any set
On the basis of this provision, combining the relevant concepts of subsets and true subsets, we can get:
An empty set has only one subset, which is itself;
An empty set is a true subset of any non-empty set
0,,
0,,3. The nature of empty sets, subsets, and true subsets
Specifies: An empty set is a subset of any set That is, for any set, there are.
Any one set is a subset of itself, ie.
If, then. If, then.
Note:An empty set is a subset of any set, so when solving a problem with parameters, it is necessary to pay attention to discussing two cases, the former is often overlooked, resulting in incomplete thinking. Here's an example:
4. The number of subsets (** in permutation and combination of knowledge points).
If there is an element in the collection, there is.
The number of subsets of is one
The number of non-empty subsets of is one
The number of true subsets of has one
The number of non-empty true subsets of has one
5. Wayne Diagram
John Venn (4 August 1834 – 4 April 1923): English mathematician In mathematics, we often use the internal representation of a set of closed curves on a flat surface, which is called a graph.
Note:
A Venn diagram representing a set is a closed curve, which can be a circle, a rectangle, an ellipse, or other closed curves;
The advantage of the Venn diagram is that the image is intuitive, and the disadvantage is that the public features are not obvious, so it is necessary to pay attention to distinguishing the relationship between size and size when drawing the diagram.
Basic operations for sets
1. Intersection
Literal language: For two given sets, the set of all the elements that belong and belong to it, is called the intersection of , denoted as, read as intersection. (In layman's terms, a new set composed of two or more common elements of a set is called a union).
Symbolic language: = and.
Graphic language: The shaded part is.
Intersection venn diagram properties: =, =, ==, if, then =
Solution: Find the same for the intersection of a single number, draw the number axis for the intersection of inequalities, and draw different sets at different heights.
Here's an example:
2. Union
Literal language: For two given sets A and B, the set consisting of all the elements of the two sets is called the union of a and b, denoted a b, and pronounced "a and b".(In layman's terms, it means that all the elements of two or more sets are put together, and the repeated elements are only taken once).
Symbolic language: a b
Graphic language: The shaded part is a b
Union venn diagram Properties: a b b a, a a a a, a a a a, if a b, then a b b
Solution: All the elements of two sets are concentrated together, but the duplicate elements are only written once, and the heterogeneity in the set must be satisfied.
Here's an example:
3. Supplement
Complete set: When studying the relationship between sets and sets, if the sets to be studied are all subsets of a given set, then the given set is called a complete set Notation: The complete set is usually denoted as u
Patch...Written Language:
If a given set A is a subset of the complete set u, the set of all elements in u that do not belong to a is called a complement in u and is denoted as .
Symbolic language: = and.
Graphic Language:
Complement the venn diagramNature:
Note:Not all complete works are represented by the letter u, and not all of them are r, depending on the title.
Here's an example:
4. The arithmetic law between setsCommutative Law:
Associative Property: Distributive Property:
De Morgan's Law: De Morgan: Mathematician, first president of the London Mathematical Society5. Use the problem solving idea of intersection and complement the parameter range
Find the parameter range according to the union:
If a has a parameter, then we need to discuss whether a is an empty set;
If b has parameters, then.
Finding the range of parameters according to the intersection:
If a has a parameter, then we need to discuss whether a is an empty set;
If b has parameters, then.
The basic relationship between sets and the value orientation of basic operations in the college entrance examination
1. Examination of basic knowledge
The basic relationship between sets and basic operations are the basic knowledge in mathematics and an important content in mathematics learning. Through the examination of the college entrance examination, students can test their mastery of these basic knowledge.
2. Examination of mathematical thinking ability
The basic relationships and operations between sets require students to have certain mathematical thinking skills, such as abstract thinking and logical thinking skills. Through the examination of these questions, it can be tested whether students have a certain mathematical thinking ability.
3. Examination of problem-solving ability
The basic relationships between sets and the problems involved in basic operations require students to have certain problem-solving skills, such as the ability to analyze and solve problems. Through the examination of these questions, it can be tested whether students have a certain ability to solve problems.
4. Examination of mathematics application ability
The basic relationships between sets and basic operations have a wide range of applications in real life, and through the examination of these problems, students can be tested to apply the mathematical knowledge they have learned to solve practical problems.
Therefore, the value orientation of the basic relationship between sets and basic operations in the college entrance examination is mainly to test students' basic knowledge, mathematical thinking ability, problem-solving ability and mathematical application ability, etc., aiming to comprehensively test students' mathematical ability and comprehensive quality.
The following are related practice questions (please bookmark if necessary).