{x 1} represents a set containing the number x 1, and [X 1, X2] represents a number greater than or equal to x 1 and less than or equal to X2.
The concept of set
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? B. If A is a subset of B and A is not equal to B, call A proper subset of B and write A? 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.
First, multiple choice questions
1. The following eight relationships ① {0} = ② = 0③ {} ④ {0} ⑤ 0⑤ {0} ⑧ {} of which the correct number ().
(A)4 (B)5 (C)6 (D)7
2. proper subset with the set {1, 2,3} has ().
Five (b) six (c) seven (d) eight
3. Let A={x} B={} C={} and then ().
(a) (a+b) A (b) (a+b) B (c) (a+b) C (d) (a+b) Any one of A, B and C.
4. Let A and B be two subsets of complete sets U and A B, then the following formula holds ().
(A)CUA cub (B)CUA cub =U
(C)A = (D)CUA B=
5. If the set A={} B={} is known, then A= ().
(A)R (B){ }
(C){ } (D){ }
6. The following statements: (1)0 and {0} represent the same set; (2) The set composed of 1, 2,3 can be expressed as {1, 2,3} or {3,2,1}; (3) The set of all solutions of equation (x- 1)2(x-2)2=0 can be expressed as {1, 1, 2}; (4) The set {} is a finite set, and the correct one is ().
(a) only (1) and (4) (B) only (2) and (3).
(c) Only (2) and (d) in the above statement are incorrect.
7. If a = {1, 2, A2-3A- 1}, B = {1, 3}, A {3, 1}, then a is equal to ().
(A)-4 or1(b)-/kloc-0 or 4 (C)- 1 (D)4
8. Let U = {0, 1, 2,3,4}, A = {0, 1, 2,3} and B = {2 2,3,4}, then (CUA) (CUB)= ().
(A){0} (B){0， 1}
(C){0， 1，4} (D){0， 1，2，3，4}
9. Let S and T be two nonempty sets, S T, t S, let X=S then S X= ().
(A)X (B)T (C) (D)S
10. Let A = {x} and B = {x}. If ab = {2,3,5}, then a and b are () respectively.
(A){3，5}、{2，3} (B){2，3}、{3，5}
(C){2，5}、{3，5} (D){3，5}、{2，5}
1 1. Let the unary quadratic equation AX2+BX+C = 0 (a
(A)R (B)
(C){ } (D){ }
(1) Product quality
Q P
(C)P=Q (D)P Q=
12. Given that P={} and Q={, the following relationship holds ().
13. If M={} and N={ Z}, then m n is equal to ().
(A) (B){ } (C){0} (D)Z
14. Among the following categories, the correct one is ().
(A)2
(B){ }
(C){ }
(D){ }={ }
15. let U={ 1, 2,3,4,5}, and a and b are subsets of u. if A B={2}, (CUA) B={4}, (CUA) (CUB)={ 1.
(A)3 (B)3
(C)3 (D)3
16. If u and represent complete sets and empty sets respectively, and (CUA) A, then sets A and B must satisfy ().
(A) (B)
(C)B= (D)A=U and A B.
17. given U=N, A={}, CUA is equal to ().
{0， 1，2，3，4，5，6}
(C){0， 1，2，3，4，5} (D){ 1，2，3，4，5}
18. The image of the quadratic function y=-3x2+mx+m+ 1 does not intersect with the X axis, so the value range of m is ().
(A){ } (B){ }
(C){ } (D){ }
19. Let the complete set U={(x, y)}, set M={(x, y)}, and N={(x, y)}, then (CUM) (CUN) is equal to ().
(A){(2，-2)} (B){(-2，2)}
(C) (D)(CUN)
20.inequality
(A){x } (B){x }
(C){ x } (D){ x }
Second, fill in the blanks
1. In rectangular coordinate system, the point set on the coordinate axis can be expressed as
2. if A={ 1, 4, x}, B={ 1, x2} and A B=B, then x=
3. if A={x} B={x} and the complete set U=R, then A=
4. If the equation 8x2+(k+ 1)x+k-7=0 has two negative roots, then the value range of k is
5. All subsets of the set {a, b, c} are proper subset yes non-empty proper subset behavior.
6. The solution set of equation x2-5x+6=0 can be expressed as
system of equations
7. Let the sets A={}, B={x}, A B, then the value range of real number K is
.
8. Let the complete set U={x is a nonnegative odd number less than 20}, if a (cub) = {3 3,7, 15}, (cua) b = {13, 17, 19},.
9. Let U={ triangle}, M={ right triangle} and N={ isosceles triangle}, then M N=
M N= cumulative value =
CUN= copper (M N)
10. Let the whole set be, and use the intersection of sets A, B and C to represent the shaded part of the graph.
( 1) (2)
(3)
Third, answer questions.
1. Let the complete set u = {1, 2,3,4} and ={ x2-5x+m=0, x U} If cua = {1, 4}, find the value of m.
2. the set A={a equation x2-ax+ 1=0 has real roots}, and B={a inequality ax2-x+1>; 0 is true for all x R}, find a B.
3. Given the set A = {A2, A+ 1, -3}, B = {A-3, 2A- 1, A2+ 1}, if A B={-3}, the real number A.
4. It is known that one of the equations x2-(k2-9)+k2-5k+6=0 is less than 1 and the other is greater than 2, so the value range of the real number k is obtained.
5. let A={x, where x R, if A B=B, the range of the real number a.
6. Let the complete set U={x}, set A={x}, B={ x2+px+ 12=0}, (CUA) B={ 1, 4,3,5}, and find the values of p and q.
7. If the inequality x2-ax+b; Solution set for 0.
8. let A={(x, y)}, let B={(x, y), and 0}, a, the range of the real number m.
The first unit set
First, multiple choice questions
The title is 1 23455 6789 10.
Answer B C B C B C B C D A
The title is11213141516171819 20.
Answer D A A D C D A D A B
Second, fill in the blanks.
1.{(x，y)} 2.0，3。 {x, or x 3} 4. {} 5. {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a. All subsets except {a, b, c}; All subsets except and {a, b, c} 6. {2,3}; {2,3} 7. {} 8. {1,5,9,11} 9. {isosceles right triangle}; {isosceles or right triangle}, {oblique triangle}, {equilateral triangle}, {neither isosceles nor right triangle}. 10. (1) (ab) (2) [(cua) (cub)]; ⑶(A B)(CUC)
Third, answer questions.
1 . m = 2×3 = 6 ^ 2。 {a } 3.a=- 1
4. hint: make F (1) < 0 and f (2) < 0.
5. Hint: A={0, -4}, A B=B, so A B.
(I) when b = 4 (a+1) 2-4 (a2-1) < 0, a is obtained.
(ii) When b = {0} or B={-4}, 0 is a=- 1.
(iii) b = {0, -4}, and a= 1 is obtained.
To sum up, the real number a= 1 or a-1.
6.U = {1, 2,3,4,5} A = {1, 4} or A = {2,3 3} Cua = {2 2,3,5} or {1,4,5} B = {
P=-(3+4)=-7 q=2×3=6
7. The solution set of equation x2-ax-b=0 is {2,3}, from Vieta theorem a=2+3=5, b=2×3=6, inequality BX2-AX+ 1 >: 0 to 6x2-5x+1>; 0 solution {x}
8. know the equation from a B.
X2+(m- 1)x=0 has a solution within 0 x, that is, m 3 or m- 1.
If it is 3, then x1+x2 =1-m.
If m- 1, x1+x2 =1-m >; 0, x 1x2= 1, so the equation has two positive roots, and both are 1 or one is greater than 1 and the other is less than 1, that is, at least one of them is in [0,2].
Therefore {m