Introduction to the knowledge points of basic relation of mathematics set in senior one, 1. 1.2 Basic relation of set.
1. Venn diagram
In mathematics, a set is represented by the interior of a closed curve on a plane, which is called a Venn diagram.
For example, the set of municipalities directly under the Central Government of China is A, which is represented by the Venn diagram as shown in the figure.
Example 1 Try to express the set A={x|x2- 16=0} by venn diagram.
Solution: Set A is the solution set of equation x2- 16=0. If the equation x2- 16=0, then x 1=4, x2=-4, so a = {-4,4} is represented by venn diagram as shown in the figure.
Understand that venn diagram and venn diagram represent an intuitive and clear set, and the closed curve can be rectangle, ellipse or circle. There is no limit.
2. Subsets
Definition Generally speaking, for two sets A and B, if any element in set A is an element in set B, we say that these two sets have an inclusion relationship, and call set A a subset of set B. notation
And the pronunciation is AB (or BA), which is pronounced as? Is A included in B? (or? B contains a? ). Explain or illustrate that people with Beijing Dongcheng hukou form set M and those with Beijing hukou form set P. Since everyone with Beijing Dongcheng hukou has Beijing hukou, there is MP. Conclusion (1) Any set is a subset of itself, namely AA.
(2) For sets A, B and C, if AB and BC, then AC. Understanding of subsets (1)? AB？ Meaning: If xA, xB can be introduced.
(2) Set A is a subset of Set B, so it cannot be understood that Set A is composed of Set B? Some elements? Because set a may be an empty set or set B.
(3) If there are elements in set A that are not set B, then set A is not included in b, or b does not include a, and it is recorded as AB or BA.
(4) Pay attention to the difference between the symbol and: it is only used between sets, such as {0}N, which cannot be written as {0} n; Can only be used between elements and collections, such as 0 N, and cannot be written as 0 n.
Example 2- 1 set M={0, 1} and set N={0, 2, 1-m}. If MN, then the real number m = _ _ _ _ _ _ _
Analysis: From the meaning of the question, we can see that MN, the set M={0, 1}, so 1N, that is, 1-m= 1. So m=0.
Answer: 0
Example 2-2 known set M={xZ|- 1? X & lt3}, N={x|x=|y|, yM}, try to judge the relationship between set m and n.
Solution: ∫xZ, and-1? x & lt3,
? The possible values of x are-1, 0, 1, 2.
? M={- 1，0， 1，2}。
∫yM is here again,
? |y| is 0, 1, 2 respectively.
? N={0， 1，2}。
? NM。
Step 3 set the equation
If set A is a subset of set B (AB) and set B is a subset of set A (BA), then set A is equal to set B, denoted as A = B, and represented by venn diagram as shown in the figure.
The understanding that sets are equal (1)A=BAB, and BA, which is an important basis to prove the equality of two sets;
(2) Set equality can also be defined from the viewpoint of elements: as long as the elements that make up two sets are the same, that is, the elements in the two sets are exactly the same, the two sets are said to be equal;
(3) The same set can have different representations, which is also the meaning of defining two sets to be equal;
(4) Analogy between set relation and real number conclusion.
Real number set a? B has two meanings: a=b, or a.
A.P={ 1，4，7}，Q={ 1，4，6}
B.P={x|2x+2=0}，Q={- 1}
C.3P，3Q
D. perceptual quantization
Analysis: For item A, 7P, but 7Q, what about P? q； For item B, p = {x | 2x+2 = 0} = {-1} = q; For item C, judging from 3p and 3q, it is uncertain whether PQ and QP are established at the same time; For item D, it is impossible to determine whether P and Q are equal only by PQ.
Answer: b
Example 3-2 Let the sets A={x, y} and B={0, x2}. If A=B, find the values of the numbers x and y.
Solution: from the definition of set equality, or.
(1) x=0, y=0, which does not satisfy the mutual anisotropy of elements in the set, so it is discarded;
(2) From x=0, y=0 or x= 1, y=0, we can know from (1) that x=0, y=0 should be discarded, and x= 1, y=0 conforms to the mutual anisotropy of elements in the set.
To sum up, x= 1, y=0.
4. proper subset
Definition If the set AB has elements xB and xA, we call the set A the proper subset of the set B, and the notation is AB (or BA). The graphic conclusion is (1)AB and BC, and then AC;
(2)AB and a? B, then AB. Understanding of proper subset (1) If set A is a subset of set B, all elements in set A belong to set B, and at least one element in set B does not belong to set A;
(2) The subset includes two situations: set equality and proper subset, and the proper subset is based on the subset. If set A is not a subset of set B, then set A must not be a proper subset of set B;
(3) Unlike any set that is a subset of itself, any set is not its own proper subset.
Example 4 given sets P={2 0 12, 2 0 13} and q = {201,2 0 12, 2 0 13, 20/kloc-3.
A.QP
C. President QP
Analysis: Obviously, all elements in set P belong to set Q, so PQ, but 2 0 14Q, 2 0 14P, so PQ.
Answer: c
5. Empty set
Definition We call a set without any elements an empty set. The symbol specifies that an empty set is a subset of any set, that is, the A-feature (1) empty set has only one subset, that is, itself.
(2) Is a proper subset of any non-empty set, that is, if a? , the difference between {0} and.
{0} and
The difference between {0} and {0} is that a set containing one element is a set without any elements, so {0}, please note that it cannot be written as ={0}. {0} Example 5- 1 The following set is empty ().
A.{0} B.{ 1}
C.{ x | x & lt0} D.{x| 1+x2=0}
Analysis: Obviously {0} and {1} are not empty sets; Because {x | x
Answer: d
Example 5-2 has the following propositions: ① An empty set has no subset; ② Any set has at least two subsets; ③ An empty set is the proper subset of any set; 4 if a, then a? Among them, the correct one is ()
A.0 B. 1 C.2 D.3
Analysis: For ①, the empty set is a subset of any set, so ① is wrong; For ②, there is only one subset, that is, itself, ② is wrong; For ③, the empty set is not the proper subset of the empty set, ③ is wrong; An empty set is the proper subset of any non-empty set, and ④ is correct.
Answer: b
6. Judge the relationship between sets
The relationship between (1) sets a and b
(2) The key to judge the relationship between two sets is to find out which elements a given set consists of, that is, to concretize the abstract set, which requires skillful use of natural language, symbolic language (enumeration and description) and graphic language (venn diagram) to represent the set.
(3) There are three main methods to judge the relationship between sets:
① List the observation results one by one;
② Characteristic method of set elements: firstly, determine what the elements of the set are, find out the characteristics of the set elements, and then use the characteristics of the set elements to judge the relationship.
Generally set A={x|p(x)} and B={x|q(x)}, if p(x) deduces q(x), then ab; If q(x) subtracts p(x), then ba; If p(x) and q(x) push each other, then A = B；; If p(x) can't deduce q(x) and q(x) can't deduce p(x), then sets A and B have no inclusion relation.
③ Number-shape combination method: using number axis or venn diagram.
(4) When both MN and MN are established, MN reflects the relationship between sets M and N more accurately than MN, and when both MN and M=N are established, M=N reflects the relationship between sets M and N more accurately than MN.
For example, the set M={ 1} and the set N={ 1, 2}. When both MN and MN are established, MN reflects the relationship between the set M={ 1} and the set N={ 1, 2} more accurately than MN. Another example is set m = {
( 1)A={- 1， 1}，B={(- 1，- 1)，(- 1， 1)，( 1)，( 1)，}；
(2)A={x|x is an equilateral triangle} and B={x|x is an isosceles triangle};
(3)A={x|- 1
(4)M={x|x=2n- 1，nN*}，N={x|x=2n+ 1，nN*}。
Analysis: first find the characteristics of elements in a set, and then judge the relationship between sets through the characteristics.
Solution: (1) The representative elements of set A are numbers, and the representative elements of set B are ordered real number pairs, so there is no inclusion relationship between A and B. 。
(2) an equilateral triangle is a triangle with three equal sides, and an isosceles triangle is a triangle with two equal sides, so AB.
(3) Set b = {x | x
(4) According to the enumeration method, M={ 1, 3,5,7,? }，N={3，5，7，9，？ }, so NM.
How to express a set with the number axis For a set composed of continuous real numbers, it is usually expressed with the number axis, which also belongs to the graphical representation of the set. Note that on the number axis, if the endpoint value is an element of the set, it is represented by a real point; If the endpoint value is not an element in the set, it is represented by a hollow point.
Example 6-2 If the set is known, then the relationship between set m and n is ().
A. MNB, Minnesota
C. nano
Analysis: let n=2m or 2m+ 1, mZ,
Then there is
.
Say it again,
? MN。
Answer: b
7. Find a subset (or proper subset) of a known set.
(1) When writing a subset of a set, you can write it in the order of the number of elements in the set from nothing to nothing and from less to more, so as to ensure that it is neither heavy nor leaking. This particular set must be considered because it is a subset of any set. If you want to write the proper subset of a set, you can't count the set itself, because any set is a subset of itself, not its proper subset.
For example, write all subsets of the set {1, 2,3} and proper subset. We can write from less to more, where the number of elements is 0, 1, 2 and 3 respectively. We can get that all subsets of the set {1, 2,3} are {1}. All proper subset are, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}.
(2) When set A contains n elements, the number of subsets is 2n, the number of proper subset is 2n- 1, the number of nonempty subsets is 2n- 1, and the number of nonempty proper subset is 2n-2.
Example 7- 1 The known set m satisfies {1, 2}M{ 1, 2,3,4,5}. Please write the set m.
Analysis: According to the conditions given in the title, the set M contains at least 1, 2, at most 1, 2, 3, 4, 5, and must contain 1, 2, so the set M can be written according to the number of elements contained in m. 。
Solution: (1) When m contains two elements, m is {1, 2};
(2) When m contains three elements, m is {1, 2,3}, {1, 2,4}, {1, 2,5};
(3) When m contains four elements, m is {1, 2,3,4}, {1, 2,3,5}, {1, 2,4,5};
(4) When m contains five elements, m is {1, 2,3,4,5}.
Therefore, the set m that meets the conditions is {1 2}, {1 2,3}, {1 2,4}, {1 2,5}, {1 2,2.
The skill of determining the subset of a finite set (1) determines the required set;
(2) Reasonable classification, writing in turn according to the number of elements contained in the subset;
(3) Pay attention to two special sets, namely the empty set and the set itself, to see if they can be retrieved.
Example 7-2 let sets A={a, b, c} and B={T|TA} and find set B. 。
Solution: A = {a, b, c}, while TA,
? T may be, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}.
? B={，{a}，{b}，{c}，{a，b}，{a，c}，{b，c}，{a，b，c}}。
Example 7-3 Given set A={ 1, 3,5}, find the sum of all the elements of set A. 。
Solution: The subsets of set A are: {1}, {3}, {5}, {1, 3}, {1, 5}, {3,5}, {1, 3,5}. Note A. 4=36.
The formula for calculating the sum of elements of all subsets of a set is if set A={a 1, a2, a3,? , an}, then the sum of the elements of all subsets of A is (a 1+a2+? +an)？ 2n- 1。
8. Basic relations of sets and synthesis of equations
The basic relationship between set and equation synthesis is usually the relationship between two sets representing the solution set of the equation, and the range of unknown parameters in the equation is found. To solve this kind of problem, we should pay attention to:
(1) It is necessary to clearly point out which letter in the set of solution sets of the equation is the unknown in the equation. The set {x|f(x)=0} represents the solution set of the equation about x, where x is unknown and other letters are constant. For example, the set {x|mx2-x+23=0} represents the equation about x, when m? 0, the equation is mx2-x+23=0, and then the quadratic equation about x 。
(2) Correctly understand the meaning of set inclusion relation, especially the meaning of AB. When b? For AB, it is usually divided into A= and a? Two cases are discussed. At this time, it is easy to ignore the case of A=.
(3) It is necessary to discuss whether the quadratic coefficient is zero for the solution set of the equation with parameters in the quadratic coefficient. Example 8- 1 If A={x|x2+x-6=0}, B={x|mx+ 1=0} and BA, find the value of m. 。
Analysis: Because BA, all elements of set B are elements of set A, but because the element X of set B satisfies mx+ 1=0, the range of the letter M is not clear, so the question whether M is 0 is not clear and needs to be classified and discussed. There are two questions to be clarified in this question: first, whether set B has elements; Second, when set B has elements, what are they?
Solution: A = {x | x2+x-6 = 0} = {-3,2}.
Because of BA, the solution of equation mx+ 1=0 can be -3 or 2 or no solution.
When the solution of mx+ 1=0 is -3, it is obtained by -3m+ 1=0;
When the solution of mx+ 1=0 is 2, it is obtained by 2m+ 1=0;
When mx+ 1=0 has no solution, m=0.
To sum up, the value of m is or or 0.
Example 8-2 Let the set A={x|x2+4x=0}, b = {x | x2+2 (a+1) x+a2-1= 0}, if BA, find the value or range of the number A. 。
Solution: A={0, -4}, BA.
(1) when A=B, B={0, -4}.
So 0, -4 is the two roots of the equation x2+2 (a+1) x+a2-1= 0.
According to Vieta's theorem, a= 1.
(2) When B=,? = 4(a+ 1)2-4(a2- 1)& lt； 0，a
(3) When b is a single element set,? = 4 (a+1) 2-4 (a2-1) = 0, and the solution is a=- 1.
When a=- 1, B={x|x2=0}={0}A, that is, the condition is met.
To sum up, the value range of real number a is a? -1 or a= 1.9. The basic relationship between sets and the synthesis of inequalities.
It is intuitive to use numbers to represent digital images, so the idea of combining numbers with shapes is widely used in mathematics. The number axis represents a real number, and any real number can be represented by a point on the number axis. Conversely, any point on the number axis represents a real number, and the range of an inequality is represented on the number axis, which is intuitive.
When representing a set on the number axis, pay attention to whether the endpoint is a solid point or a hollow point. If the endpoint is included, it is indicated by a solid line point; If not, it is indicated by a hollow dot.
The basic relationship between set and inequality synthesis is usually that the relationship between two sets of inequality solutions is known, and the values (or ranges) of parameters in inequality are required. To solve this kind of problem, we should pay attention to:
(1) Make it clear which letter in the set of inequality solutions is the unknown of inequality. set { x | f(x)>； 0}，{ x | f(x)& lt； 0}，{x|f(x)？ 0}，{x|f(x)？ 0} all represent the solution set of inequality about x, where x is unknown and other letters are constant. For example, the set {x |-NX+3
(2) Formulas connected by unequal signs are called inequalities, such as 2.
Analysis: Set A is an inequality represented by a specific number, set B is an inequality represented by the letter M, and the inequality given by set A is a line segment from -2 to 5 on the number axis (except for the two endpoints). There are two cases of inequality given by set B: when m+ 1 and 2m- 1? 2m- 1 x, so BA must be established; When m+ 1
Solution: ∵BA, a? ,? B= or b? .
When B=, m+ 1? 2m- 1，m？ 2.
b？ As shown on the number axis.
Then there is a solution.
sulphur dioxide
To sum up, the value range of m is m? 2 or 2
Example 9-2 Known Set a = {x | x
Analysis: Discuss whether the solution set B is empty.
Solution: when B=, only 2a >; A+3, namely a>3;
When b? According to the meaning of the question, make the number axis as shown in the figure.
A
To sum up, the range of real number A is A.
When using subset relation to find parameters, it is easy to ignore the verification of endpoints. When using subset relation to find parameters, we should pay attention to verifying whether the parameters can get endpoint values. Like in b? A