The knowledge in this chapter is mainly divided into three parts: set, simple inequality solution (set simplification) and simple logic:

Second, knowledge review:

(1) assembly

1. Basic concepts: set and element; Finite set, infinite set; Empty set and complete set; The use of symbols.

2. Representation of sets: enumeration, description and graphic representation.

3. The characteristics of set elements: certainty, mutual difference and disorder.

4. Set operation: intersection, union and complement.

5. Main characteristics and operation rules

(1) Inclusion relation:

(2) Equivalence relation:

(3) Set algorithm:

Exchange method:

Association rule:

Distribution law:.

0- 1 Law:

Equal rights law:

The law of complement: A∩? UA =φA∨？ UA=U？ UU=φ？ Uφ=U？ UU(？ UA)=A

Inverse law:? U(A∩B)=(？ UA)∩(？ UB)？ U(A∪B)=(？ UA)∩(？ UB)

6. The number of elements in a finite set

Definition: The number of elements in a finite set A is called the cardinality of set A, which is denoted as card( A), and it is stipulated that card(φ) =0.

Basic formula:

3 card (? UA)= card (U)- card (a)

(4) Let finite set A and card (a) = n, then

(i) The number of subsets of a is; (ii) The proper subset number of a is;

(iii) The number of nonempty subsets of A is; (iv) The number of nonempty proper subset of A is.

(5) Let the finite sets A, B, C, Card (A)=n, Card (b) = m, M.

(i) If, the number of c is;

(ii) If yes, the number of c is;

(iii) If, the number of c is;

(iv) If, then the number of c is.

(2) The solution and generalization of absolute inequality and unary quadratic inequality.

Solution of 1. Algebraic expression inequality

Root axis method (zero division method)

① transform the inequality into A0 (x-x1) (x-x2) ... (x-XM) > 0 (< 0) form, the coefficient of each factor x is "+"; (For the sake of uniformity and convenience)

② Find the root, which is expressed on the number axis;

(3) Starting from the upper right thread, each thread is represented by the number of points on the shaft (why? );

(4) If the inequality (after the coefficient x is "+") is ">; 0 ",and then find the interval of" line "above the X axis; If the inequality is "

(right-to-left plus sign and minus sign)

Then the solution of inequality can be determined according to the symbol of each interval.

Special case ① One-dimensional linear inequality ax> discusses the best solution;

② unary quadratic inequality ax2+box & gt；; 0(a & gt； 0) Discussion of solutions.

quadratic function

Image of ()

monadic quadratic equation

There are two different real roots.

There are two equal real roots.

There is no real root

rare

2. Solving the Fractional Inequality

(1) Standardization: Empathy is divided into >: 0 (or

(2) Inequalities transformed into algebraic expressions (groups)

3. Solve inequalities with absolute values

(1) formula method: the solution of sum inequality.

(2) Definition method: use the "zero division method" for classification discussion.

(3) Geometric method: According to the geometric meaning of absolute value, solve the problem by combining numbers and shapes.

4. Distribution of roots of quadratic equation with one variable

The unary quadratic equation ax2+bx+c=0(a≠0)

"Zero distribution" of (1) root: analysis and solution according to discriminant and Vieta theorem.

(2) The "non-zero distribution" of roots: make a quadratic function image and solve it by combining numbers and shapes.

(3) Simple logic

1. Definition of a proposition: A statement that can be judged to be true or false is called a proposition.

2. Logical connectives, simple propositions and compound propositions:

Words such as "or" and "not" are called logical conjunctions; A proposition without logical conjunctions is a simple proposition; A proposition composed of simple propositions and logical conjunctions "OR" and "NOT" is a compound proposition.

Form of compound proposition: P or Q (marked as "P ∨ Q"); P and q (marked "p ∧ q"); Non-P (recorded as "┑q").

3. Truth judgment of "OR", "AND" and "NOT"

The truth value of (1) "non-p" compound proposition is opposite to f;

(2) The compound proposition in the form of "P and Q" is true when both P and Q are true, and false in other cases;

(3) The compound proposition in the form of "P or Q" is false when both P and Q are false, and true in other cases.

4. The form of four propositions:

Original proposition: if p is q; Inverse proposition: if q is p;

There is no proposition: if ┑P is ┑ Q; Negative proposition: If ┑q, then ┑ p

(1) exchange the conditions and conclusions of the original proposition, and the obtained proposition is an inverse proposition;

(2) Denying the conditions and conclusions of the original proposition at the same time, and whether the obtained proposition is a proposition;

(3) Exchange the conditions and conclusions of the original proposition and deny it at the same time, and the obtained proposition is a negative proposition.

5, the relationship between the four propositions:

There are three relationships between the truth value of one proposition and the truth values of the other three propositions: (the original proposition is negative)

(1), the original proposition is true, but its inverse proposition is not necessarily true.

② The original proposition is true, but its negative proposition is not necessarily true.

③ If the original proposition is true, its negative proposition must be true.

6. if p q is known, then we say that p is a sufficient condition of q and q is a necessary condition of p.

If p q, q p, then p is the necessary and sufficient condition of q, which is recorded as p? Ask.

7. Reduction to absurdity: Starting from the opposite side of the conclusion of a proposition (hypothesis), it leads to contradictions (with known, axioms, theorems, etc. ), thus denying the hypothesis and proving the original proposition. This method of proof is called reduction to absurdity.