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

Description

"Sufficient Conditions and 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 necessary and sufficient evidence

General approach to conditions

Sufficient and necessary conditions PPT, part 2: independent learning

Problem guide

Preview textbooks P17-P22 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

The proposition is true or false. "If p, then q" is a true proposition. "If p, then q" is a false proposition.

Deduced relation P____q p____ q

Conditional relationship p is a _______ condition of q p is not a ______ condition of q

q is a _______ condition of p q is not a ______ condition of p

■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 "if p, then q" and its converse "if q, then p" are both true propositions, that is, there are both ________ and ________, then it is recorded as ________. At this time, p is both a sufficient condition for q 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 conditions 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

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

Judgment of sufficiency, necessity, and necessary and sufficient conditions

In each of the following questions, what condition does p represent for 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.

[Solution] (1) Because x=1 or x=2⇒x-1=x-1, x-1=x-1⇒x=1 or x=2, so p is a necessary and sufficient condition of q.

(2) If a quadrilateral is a square, then its diagonals bisect each other perpendicularly, that is, p⇒q. On the contrary, if the diagonals of a quadrilateral bisect each other perpendicularly, the quadrilateral is not necessarily a square, that is, q⇒/p.

So p is a sufficient and unnecessary condition for q.

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 condition 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

Sufficient 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 and unnecessary conditions

B． Necessary and insufficient conditions

C. Necessary and sufficient conditions

D. Neither sufficient nor necessary conditions

Analysis: Choose B. From the congruence of two triangles, it can be concluded that the areas of the two triangles are equal. The opposite is not true.

Therefore, "the areas of two triangles are equal" is a necessary and insufficient condition for "two triangles are congruent". Therefore choose B.

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

Analysis: Choose A. When a=1, N={1}, then N⊆M; when N⊆M, a2=1 or a2=2, the solution is a=1 or -1 or 2 or -2 .So "a=1" is a sufficient and unnecessary condition for "N⊆M".

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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Update Time: 2024-08-06

This template belongs to Mathematics courseware People's Education High School Mathematics Edition A Compulsory Course 1 industry PPT template

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