First, the curriculum standard requirements:
We can judge sufficient conditions, necessary conditions and necessary and sufficient conditions by understanding their meanings.
Second, review knowledge and methods:
1, the concepts of sufficient condition, necessary condition and necessary and sufficient condition;
2. From the perspective of logical reasoning, sufficient and unnecessary conditions, necessary and insufficient conditions, necessary and sufficient conditions:
3. Sufficient conditions, necessary conditions and necessary and sufficient conditions are viewed from the relationship between sets:
4. Special value method: When judging sufficient conditions and necessary conditions, the special value method is often used to deny the conclusion.
5, return to thought:
It means that P and Q are equivalent, and equivalent propositions can be transformed into each other. When we want to prove that P is true, it can be transformed into proving that Q is true.
It should be noted here that the original proposition, negative proposition and negative proposition are only one of the equivalent forms, and the idea of reduction is generally applicable to propositions whose conditions or conclusions are inequality relations (negative expressions).
6. Combination of numbers and shapes:
Judging sufficient and unnecessary conditions, necessary and insufficient conditions, necessary and sufficient conditions by using Wayne diagram (that is, the inclusion relationship of sets).
Third, basic training:
1. Let the proposition be that if P is false, if Q is true, then P is Q's ().
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
2. Let the sets M and N be two subsets of the complete set U, then it is ()
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
3. If it is a real number, it is ().
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
Fourth, give an example
Example 1 A quadratic equation with real coefficients is known, and the following conclusion is correct ().
(1) is a necessary and sufficient condition for this equation to have a real root.
(2) It is a necessary and sufficient condition that this equation has a real root.
(3) It is a necessary and sufficient condition that this equation has real roots.
(4) It is a necessary and sufficient condition that this equation has real roots.
A.( 1)(3)b .(3)(4)c .( 1)(3)(4)d .(2)(3)(4)
Example 2 (1) Given h 0, A, bR, let Proposition A: and Proposition B: and find A as B ().
(2) It is known that p: the slopes of two straight lines are negative reciprocal, and q: the two straight lines are perpendicular to each other, so p is q ().
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
Variant: a = 0 is the condition of straight line and parallelism;
Example 3 If both propositions P and Q are necessary conditions for proposition R, then proposition S is a sufficient condition for proposition R, and proposition Q is proposition S..
Sufficient conditions, then proposition P is the condition of proposition Q; Proposition s is the condition of proposition q; Proposition r is the condition of proposition Q.
Example 4 let the proposition P: | 4x-3 | 1 and the proposition q: x2-(2a+1) x+a (a+1) 0. If ﹁p is the necessary and sufficient condition of ﹁q, find the range of the number A;
Example 5 Let it be two real roots of the equation, and try to analyze the condition that both real roots are greater than 1. And give proof.
Verb (abbreviation of verb) classroom practice
1, let proposition p: and proposition q:, then p is () of q.
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
2. Give the following four propositions: ① If P is Q 2, if R is Q 3, if R is S.
(4) If ﹁s is q, and if they are all true propositions, then ﹁p is the condition of s;
3. Is there a sufficient condition for the real number P to be true? If it exists, find out the range of p; If it does not exist, explain why.
Sixth, the class summary:
Seven, teaching postscript:
Name and Date of Class 3 Student Number in Senior High School: Month Day
1, A B is AB=B ()
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
2. Yes ()
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
3, 2x2-5x-30 is a necessary and sufficient condition ()
A.-
4, 2 and b are () of a+b4 and ab.
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
5. Let a 1, b 1, c 1, a2, b2 and c2 all be nonzero real numbers, and the inequality A1x2+B1c, A2x2+B2x.
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Sufficient and necessary conditions D. Conditions that are neither sufficient nor necessary
6. If Proposition A: and Proposition B:, then Proposition A is the condition of B;
7. If the condition P: | x | = x and the condition Q: x2-x, then P is the condition of Q;
8. Equation mx2+2x+ 1=0 has at least one negative root if and only if;
9. The equation x2+mx+n = 0 of x has two positive roots less than 1 if and only if:
10, known and verified: if and only if;
1 1, P:-2 10, Q: 1-M 1+M, if ﹁p is the necessary and sufficient condition of ﹁q, find the range of the number m.
12, given the equation about X (1-a)x2+(a+2)x-4=0, aR, find:
Necessary and sufficient conditions for (1) equation to have two positive roots;
(2) Necessary and sufficient conditions for the equation to have at least one positive root.