I. Collection of related concepts

The meaning of 1. set

2. Three characteristics of elements in a set:

The certainty of (1) element is as follows: the highest mountain 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 of sets: enumeration and description.

U 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

1) enumeration: {A, b, C...}

2) Description: A method of describing the common attributes of elements in a collection and writing them in braces to represent the collection. {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 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, which is marked as AB or BA.

2. "Equality" relationship: A=B (5≥5 and 5≤5, then 5=5)

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. Answer? A

② proper subset: If a? B and a? B then says that set A is the proper subset of set B, and it is denoted as AB (or BA).

3 if a? B，B？ C, then a? C

4 if a? 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.

U has a set of n elements, including 2n subsets and 2n- 1 proper subset.

First, the common solution of function domain:

1, the denominator of the fraction is not equal to zero; 2. The number of even roots is greater than or equal to zero; 3. The real number of logarithm is greater than zero; 4. The bases of exponential function and logarithmic function are greater than zero and not equal to1; 5. Tangent function in trigonometric function; In cotangent function; 6. If the function is an analytical formula determined by the actual meaning, its value range should be determined according to the actual meaning of the independent variable.

Second, resolution function's common solution:

1, define the method; 2. Alternative methods; 3. undetermined coefficient method; 4. Function equation method; 5. Parameter method; 6. Matching method

Three, the common solution of function range:

1, substitution method; 2. Matching method; 3. Discrimination method; 4. Geometric method; 5. Inequality method; 6. Monotonicity method; 7. Direct instruction

Four, the common methods to find the maximum value of the function:

1, matching method; 2. Alternative methods; 3. Inequality method; 4. Geometric method; 5. Monotonicity method

Five, the common conclusion of monotonicity of function:

1, if they are all increase (decrease) functions in an interval, they are also increase (decrease) functions in this interval.

2. If it is an increase (decrease) function, it is a decrease (increase) function.

3. If it has the same monotonicity as, it is increasing function; If it is different from monotonicity, it is a decreasing function.

4. Odd functions have the same monotonicity in symmetric intervals, while even functions have the opposite monotonicity in symmetric intervals.

5. Monotonicity solution of common functions: size comparison, domain evaluation, maximum value, inequality solution, inequality proof and function image making.

Six, the common conclusion of function parity:

1. If a odd function is defined, then if a function is both a odd function and an even function, then (the opposite is not true).

2. The sum (difference) of two odd (even) functions is an odd (even) function; The product (quotient) of is an even function.

3. The product (quotient) of odd function and even function is odd function.

4. A function composed of two functions, as long as one of them is an even function, the composite function is an even function; When both functions are odd function, the composite function is odd function.

5. If the domain of the function is symmetrical about the origin, it can be expressed as: the right end of this formula is the sum of a odd function and an even function.