The first chapter is the concept of set and function.
I. Collection of related concepts
1, meaning of set: some specified objects are set together into a set, where each object is called an element.
2. Three characteristics of elements in a set:
1. element determinism; 2. Mutual anisotropy of elements; 3. The 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 set, it has only one 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. Representation of assembly: {…} such as {basketball players in our school}, {Pacific Ocean, Atlantic Ocean, Indian Ocean, Arctic Ocean}
1.Set is expressed in Latin letters: A={ basketball player of our school}, B={ 1, 2, 3, 4, 5}
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 "belonging"
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, marked as A ∈ A; On the other hand, if a does not belong to the set a, it is marked 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. A method to indicate whether some objects belong to this set under certain conditions.
① Language Description: Example: {A triangle that is not a right triangle}
② Description of mathematical formula: Example: inequality X-3 >; The solution set of 2 is {x? r | x-3 & gt； 2} or {x | x-3 >；; 2}
4, 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}
Second, the basic relationship between sets
1. "Inclusive" relation-subset
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 included in set B, or set B does not include set A, so it is recorded as A B or B A.
2. "Equality" relationship (5≥5, and 5≤5, then 5=5)
Example: let a = {x | x2-1= 0} b = {-1,1} "The elements are 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.
② proper subset: If AíB and A 1 B, then set A is the proper subset of set B, and it is denoted as A B (or B A).
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.
Third, the operation of the set.
Definition of 1. intersection: Generally speaking, the set consisting of all elements belonging to A and B is called the intersection of A and B. 。
Write A∩B (pronounced "A to B"), that is, A∩B={x|x∈A, x ∈ b}.
2. Definition of union: Generally speaking, 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 A∪B (pronounced as "A and B"), that is, A∪B={x|x∈A, or x ∈ b}.
3. The nature of intersection and union: A∩A = A, A∪φ=φ, A∪B = B∪A, A∪A = A,
A∪φ= A，A∪B = B∪A。
4. 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}
S
CsA
A
(2) Complete set: If the set S contains all the elements of each set we want to study, it can be regarded as a complete set, usually expressed by U. 。
(3) Properties: (1) cu (cua) = a2 (cua) ∩ a = φ 3 (cua) ∪ a = u.
Second, the related concepts of function
The concept of 1. function: Let a and b be non-empty number sets. If any number X in set A has a unique number f(x) corresponding to it according to a certain correspondence F, then F: A → B is called a function from set A to set B, and is denoted as y = f (x). The value of y corresponding to the value of x is called the function value, and the set of function values {f(x)| x∈A} is called the range of the function.
Note: 2 If only the analytical formula y=f(x) is given without specifying its domain, the domain of the function refers to the set of real numbers that can make this formula meaningful; The definition and range of functions should be written in the form of sets or intervals.
Domain supplement
The set of real numbers x that can make a function meaningful is called the domain of the function. The main basis for finding the domain of function is that the denominator of (1) score is not equal to zero; (2) The number of even roots is not less than zero; (3) The truth value of the logarithmic formula must be greater than zero; (4) Exponential radix and logarithmic radix must be greater than zero and not equal to 1. (5) If a function is composed of some basic functions through four operations, then its domain is a set of values of x that make all parts meaningful. (6) Exponential radix cannot be equal to zero. (6) The definition domain of function in practical problems should also ensure that practical problems are meaningful.
(Also note that finding the solution set of inequality group is the domain of function. )
The three elements of a function: definition domain, correspondence relationship and value domain.
Note again: (1) The three elements that make up a function are domain, correspondence and value. Because the range is determined by the domain and the corresponding relationship, two functions are called equal (or the same function) if and only if their domain and the corresponding relationship are exactly the same, but the independent variables and function values are represented by letters. (2) Domain consistency (two points must be met at the same time)
(See related example 2 on page 2 1 of the textbook)
Value range supplement
(1), the range of a function depends on the defined range and the corresponding law. No matter what method is adopted to find the range of a function, the defined range should be considered first. (2) The range of linear function, quadratic function, exponential function, logarithmic function and trigonometric function should be familiar, which is the basis for solving the range of complex variable function.
3. Function image knowledge induction
(1) Definition: In the plane rectangular coordinate system, the set c of points P(x, y) with functions y = f (x) and (x ∈ a) as abscissa and function y as ordinate is called the image of functions y = f (x) and (x ∈ a).
The coordinates (x, y) of each point on c satisfy the functional relationship y=f(x). On the other hand, the points (x, y) whose coordinates are x and y for each group of ordered real numbers satisfying y=f(x) are all on c, that is, c = {p (x, y) | y = f (x).
Image c is generally a smooth and continuous curve (or straight line), or it may be composed of several curves or discrete points, and it has at most one intersection with any straight line parallel to the Y axis.
(2) Painting
A. Point tracing method: according to the resolution function and the definition domain, find some corresponding values of x and y and list them, trace the corresponding points p (x, y) in the coordinate system with (x, y) as coordinates, and finally connect these points with smooth curves.
B, image transformation method (please refer to the compulsory 4 trigonometric function)
There are three commonly used transformation methods, namely translation transformation, expansion transformation and symmetry transformation.
(3) Function:
1, intuitively see the nature of the function; 2. Analyze the thinking of solving problems by combining numbers and shapes to improve the speed of solving problems.
Find mistakes in solving problems.
4. Understand the concept of interval.
Classification of (1) interval: open interval, closed interval and semi-open and semi-closed interval; (2) Infinite interval; (3) The number axis representation of the interval.
5. What is mapping?
Generally speaking, let A and B be two nonempty sets. If any element X in set A has a unique element Y corresponding to it according to a certain corresponding rule F, then the corresponding F: A B is the mapping from set A to set B, which is called "F: A B".
Given a mapping from set a to set b, if A ∈ A, B ∈ B and element a correspond to element b, then we call element b the image of element a and element a the original image of element B.
Note: Function is a special mapping, and mapping is a special correspondence. ① Set A, B and corresponding rule F are definite; (2) The correspondence rule is directional, that is, it emphasizes the correspondence from set A to set B, which is generally different from the correspondence from b to a; ③ For mapping F: A → B, it should be satisfied that: (i) every element in set A has an image in set B, and the image is unique; (ii) Different elements in set A and corresponding images in set B can be the same; (Ⅲ) Every element in set B is not required to have an original image in set A. 。
Common function representations and their respective advantages;
The 1 function image can be a continuous curve, a straight line, a broken line, a discrete point, etc. Pay attention to the basis of judging whether a graph is a function image; 2 analytical method: the domain of the function must be specified; 3 mirror image method: attention should be paid to drawing by tracing point method: determine the definition domain of function; Simplify the analytical formula of the function; Observe the characteristics of the function; List method: the selected independent variables should be representative and reflect the characteristics of the field.
Note: analytical method: it is easy to calculate the function value. List method: it is easy to find out the function value. Mirror image method: convenient to measure function value.
Supplement 1: piecewise function (see textbook P24-25)
There are different analytic expression functions in different parts of the domain. When finding the function value in different ranges, the independent variable must be substituted into the corresponding expression. The analytic expression of piecewise function cannot be written as several different equations, but several different expressions of function value are written as a left bracket, which respectively represents the independent variable values of each part. (1) piecewise function is one function, so don't mistake it for several functions. (2) The definition domain of piecewise function is the union of the definition domain of each segment, and the value domain is the union of the value domain of each segment.
Supplement 2: Composite Function
If y=f(u), (u∈M), u=g(x), (x∈A), then y=f[g(x)]=F(x), (x∈A) is called the composite function of f and g. 。
For example: y=2sinX y=2cos(X2+ 1)
7. Monotonicity of functions
(1). Incremental function
Let the domain of function y=f(x) be I, if any two independent variables x 1 and X2 are in an interval d within the domain I, when X 1