"Sufficient Conditions, Necessary Conditions" collection and common logical terms PPT courseware

"Sufficient Conditions, Necessary Conditions" collection and common logical terms PPT courseware

Part One: Learning Objectives

Understand the concepts of sufficient conditions, necessary conditions, and necessary and sufficient conditions

Master the method of judging sufficient conditions, necessary conditions, and necessary and sufficient conditions based on specific propositions

Master the general methods of proving necessary and sufficient conditions

Sufficient conditions and necessary conditions PPT, part 2: independent learning

Problem guide

Preview textbooks P30-P34 and think about the following questions:

1. What are sufficient conditions?

2. What are the necessary conditions?

3. What are necessary and sufficient conditions?

A preliminary exploration of new knowledge

1. Sufficient and necessary conditions

■Instructions from famous teachers

For "p⇒q", it contains the following multiple interpretations

(1) A proposition of the form "if p, then q" is a true proposition.

(2) The conclusion q can be obtained from the condition p.

(3) p is a sufficient condition for q or the sufficient condition for q is p.

(4) As long as there is a condition p, there must be a conclusion q, that is, p is sufficient for q.

(5)q is a necessary condition for p or a necessary condition for p is q.

(6) In order to obtain the conclusion q, it can be deduced if the condition p is met.

Obviously, "p is a sufficient condition for q" and "q is a necessary condition for p" express the same logical relationship, that is, p⇒q, but they are expressed in different ways.

[Reminder] "If p, then q" cannot be confused with "p⇒q". Only when "if p, then q" is a true proposition, can there be "p⇒q", that is, "p ⇒q”⇔“If p, then q” is a true proposition.

2. Necessary and Sufficient Condition

If ________, and ________, write it as ________. At this time, p is both a sufficient condition and a necessary condition for q. We say that p is a ___________ condition for q, which is referred to as a necessary and sufficient condition.

■Instructions from famous teachers

(1) p is a necessary and sufficient condition for q, which means "if p is true, then q must be true; if p is not true, then q must not be true".

(2) To determine whether p is a necessary and sufficient condition for q, two judgments are required: one is to see whether p can deduce q, and the other is to see whether q can deduce p. If p can deduce q, q can also deduce p. It can be said that p is a necessary and sufficient condition for q. Otherwise, it cannot be said that p is a necessary and sufficient condition for q.

self-test

Judge whether it is true or false (mark “√” if it is correct and “×” if it is wrong)

(1) "x=0" is a sufficient and unnecessary condition for "(2x-1)x=0". ()

(2) When q is a necessary condition for p, p is a sufficient condition for q. ()

(3) If p is a necessary and sufficient condition for q, then propositions p and q are two mutually equivalent propositions. ()

(4) When q is not a necessary condition for p, “p⇒/q” holds. ()

Let p: "The quadrilateral is a rhombus", q: "The diagonals of the quadrilateral are perpendicular to each other", then p is the () of q

A． Sufficient and unnecessary conditions

B. Necessary and insufficient conditions

C． Necessary and Sufficient Condition

D. Neither sufficient nor necessary conditions

Assume p: x<3, q: -1

A． Necessary and sufficient conditions

B. Sufficient and unnecessary conditions

C． Necessary and insufficient conditions

D. Neither sufficient nor necessary conditions

Assume a and b are real numbers, then "a+b>0" is () of "ab>0"

A． Sufficient and unnecessary conditions

B. Necessary and insufficient conditions

C． Necessary and sufficient conditions

D. Neither sufficient nor necessary conditions

Sufficient conditions and necessary conditions PPT, the third part: lecture, practice and interaction

Judgment of sufficiency, necessity, and necessary and sufficient conditions

In each of the following propositions, what is the condition for p to be q? (referring to conditions that are sufficient and unnecessary, necessary but not sufficient, sufficient and necessary, neither sufficient nor necessary)

(1)p: x=1 or x=2, q: x-1=x-1;

(2)p: The quadrilateral is a square, q: The diagonals of the quadrilateral bisect each other perpendicularly;

(3)p: xy>0, q: x>0, y>0.

(4)p: The diagonals of a quadrilateral are equal, q: The quadrilateral is a parallelogram.

regular method

Judgment method of sufficient, necessary and necessary and sufficient conditions

(1)Definition method

If p⇒q, q⇒/p, then p is a sufficient and unnecessary condition of q;

If p⇒/q, q⇒p, then p is a necessary and insufficient condition of q;

If p⇒q, q⇒p, then p is a necessary and sufficient condition for q;

If p⇒/q and q⇒/p, then p is neither a sufficient nor a necessary condition for q.

(2)Collection method

For the set A={x|x satisfies condition p}, B={x|x satisfies condition q}, the specific situation is as follows:

If A⊆B, then p is a sufficient condition for q;

If A⊇B, then p is a necessary condition for q;

If A=B, then p is a necessary and sufficient condition for q;

If A�B, then p is a sufficient and unnecessary condition of q;

If A�B, then p is a necessary and insufficient condition of q.

Track training

1. (2019•Chaozhou Final) It is known that the proposition p: -1

A． Sufficient and unnecessary conditions B. Necessary and insufficient conditions

C． Necessary and sufficient conditions D. Neither sufficient nor necessary conditions

2. (2019•Jinhua Final) "x>a" is () of "x>|a|"

A． Sufficient and unnecessary conditions B. Necessary and insufficient conditions

C． Necessary and sufficient conditions D. Neither sufficient nor necessary conditions

Proof of necessary and sufficient conditions

Verify: The necessary and sufficient condition for the quadratic equation ax2+bx+c=0 to have one positive root and one negative root is ac<0.

regular method

Proof ideas of necessary and sufficient conditions

(1) According to the definition of necessary and sufficient conditions, when proving necessary and sufficient conditions, one must prove them from two aspects: sufficiency and necessity:

Generally, it is proved that "the necessary and sufficient condition for p to be established is q";

①Sufficiency: Treat q as a known condition, combine the preconditions of the proposition, and deduce p;

②Necessity: Treat p as a known condition, combine the preconditions of the proposition, and deduce q.

The key to solving the problem is to distinguish which is the condition and which is the conclusion, and then determine the direction of deduction. There is no hard requirement as to whether to prove sufficiency or necessity first.

(2) In the proof process, if we can ensure that each step of reasoning has equivalence (⇔), we can also directly prove the necessity and sufficiency.

Sufficient conditions and necessary conditions PPT, Part 4: Feedback on compliance with standards

1. "The areas of two triangles are equal" is () of "the areas of two triangles are congruent"

A． Sufficient but unnecessary condition B. necessary but not sufficient conditions

C． Necessary and sufficient conditions D. Neither sufficient nor necessary conditions

2. Suppose the set M={1, 2}, N={a2}, then "a=1" is () of "N⊆M"

A． Sufficient and unnecessary conditions

B. Necessary and insufficient conditions

C． Necessary and Sufficient Condition

D. Neither sufficient nor necessary conditions

3. (2019•Foshan Testing) It is known that p: "x=2", q: "x-2=2-x", then p is () of q

A． Sufficient and unnecessary conditions B. Necessary and insufficient conditions

C． Necessary and sufficient conditions D. Neither sufficient nor necessary conditions

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For more information about the PPT courseware "Sets and Common Logic Terms Sufficient Conditions and Necessary Conditions", please click the Sets and Common Logic Terms Sufficient Conditions and Necessary Conditions ppt tag.

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