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

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

Part One: Explanation of Curriculum Standards

1. Understand the relationship between true propositions and derived symbols, and appreciate the superiority of symbolic language.

2. Understand the concepts of sufficient conditions, necessary conditions, and necessary and sufficient conditions, and master the judgment methods of sufficient conditions, necessary conditions, and necessary and sufficient conditions.

3. Master the general methods of proving necessary and sufficient conditions.

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

1. Sufficient conditions and necessary conditions

1. (1) It is known that "if p, then q" is a true proposition, what is the relationship between p and q?

Tip: Explain that when p is established, it must be concluded that q is established. That is, q can be obtained through reasoning from p. At this time, we say that q can be derived from p, which is recorded as p⇒q.

(2) Similarly, if "if p, then q" is a false proposition, what is the relationship between p and q?

Tip: Show that the conclusion q cannot be derived from the condition p, denoted as p q.

(3) Observe the following circuit diagram. Condition p: "Switch A is closed", conclusion q: "Light bulb B is on". When switch A is closed, will bulb B definitely light up? What does it mean? If "Light bulb B does not light up", "Can switch A be closed"?

Tip: It will definitely light up. To make "bulb B light up", it is enough to have "switch A closed".

If "bulb B does not light up", then switch A must not be closed.

(4) In the circuit below, condition p: "switch A is closed" is true, and conclusion q: "bulb B is on" is true?

Tip: It is not true. In other words, "if p, then q" is a false proposition.

2. Fill in the blanks

Generally speaking, if the proposition "if p, then q" is true, it is said that p is a sufficient condition for q and q is a necessary condition for p.

3. Do it

Fill in the blanks with "sufficient conditions" and "necessary conditions":

(1) If p:x=-3, q:x2=9, then p is the ________ of q, and q is the ________ of p.

(2) If p: the areas of the two triangles are equal, and q: the areas of the two triangles are congruent, then p is the ________ of q, and q is the ________ of p.

Answer: (1) Sufficient conditions, necessary conditions (2) Necessary conditions, sufficient conditions

2. Necessary and sufficient conditions

1. (1) We know that when "x>1" is established, we can deduce "x>0". So, can the sufficient condition of "x>0" only be "x>1"?

Tip: No. The conditions for the conclusion "x>0" to be true are not unique, such as "x>1.2", "3

(2) From the previous knowledge, we know that "x>0" is a necessary condition for "x>1". So, can the necessary condition for "x>1" only be "x>0"?

Tip: No. For example, "x>1" can also deduce "x>-1", "x≥", etc. These are the necessary conditions for "x>1" to be established.

(3) Given the condition p: "The triangle is an equilateral triangle", the conclusion q: "The three sides of the triangle are equal", then what condition is p for q? What condition is for q to be p?

Tip: p⇒q, q⇒p. p is a sufficient condition for q, q is a sufficient condition for p, p is a necessary condition for q, and q is also a necessary condition for p.

Sufficient conditions and necessary conditions PPT, the third part: analysis of example questions

Explore the judgment of sufficient conditions, necessary conditions and necessary and sufficient conditions

Example 1 (1) For any x, y∈R, "xy=0" is () of "x2+y2=0"

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

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

(2) Suppose the two diagonals of the quadrilateral ABCD are AC and BD, then "the quadrilateral ABCD is a rhombus" is "AC⊥BD" ()

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

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

(3) Suppose A and B are two sets, then "A∩B=A" is "A⊆B" ()

A. Sufficient and unnecessary conditions

B. Necessary and insufficient conditions

C. Sufficient and necessary conditions

D.Neither sufficient nor necessary conditions

Analysis: (1) From x2+y2=0, we get x=0 and y=0,

From xy=0, we get x=0 or y=0, that is, "xy=0" "x2+y2=0".

(2) If "quadrilateral ABCD is a rhombus", obviously the diagonals are vertical;

But "AC⊥BD" cannot deduce "the quadrilateral ABCD is a rhombus", such as an isosceles trapezoid with vertical diagonals.

Therefore, "the quadrilateral ABCD is a rhombus" is a sufficient and unnecessary condition for "AC⊥BD".

(3)∵A∩B=A⇔A⊆B, ∴ “A∩B=A” is a necessary and sufficient condition for “A⊆B”.

Answer:(1)A　(2)A　(3)C

Extended exploration In Example 1(2), the "quadrilateral ABCD" in the original condition is changed to "parallelogram ABCD", and the rest remains unchanged. Does the conclusion change?

Solution: If the condition is a parallelogram, then "ABCD is a rhombus" is a necessary and sufficient condition for "AC⊥BD".

Variation training 1 Let A and B be two mutually different sets. Proposition p: x∈A∩B; proposition q: x∈A or x∈B. Then p is the () condition of q.

A. Sufficient and necessary B. Sufficient and unnecessary

C. Necessary and insufficient D. Neither sufficient nor necessary

Analysis: If the proposition p: x∈A∩B is true, the proposition q: x∈A or x∈B must be true; if the proposition q: x∈A or x∈B is true, but x is not necessarily an element in A∩B , so p is a sufficient and unnecessary condition for q.

Answer:B

Sufficient conditions and necessary conditions PPT, Part 4: Class drills

1. "a=-3" is () of "|a|=3"

A. Sufficient and unnecessary conditions

B. Necessary and insufficient conditions

C. Sufficient and necessary conditions

D.Neither sufficient nor necessary conditions

Answer:A

2. "x>2" is () of "x>1"

A. Sufficient and unnecessary conditions

B. Necessary and insufficient conditions

C. Sufficient and necessary conditions

D.Neither sufficient nor necessary conditions

Answer:A

3. It is known that a and b are real numbers, then "a>0 and b>0" is the ________ condition of "a+b>0 and ab>0".

Analysis: a>0 and b>0⇒a+b>0 and ab>0; a+b>0 and ab>0⇒a>0 and b>0, so it is a necessary and sufficient condition.

Answer: sufficient and necessary

4. 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.

Proof: sufficiency:

Because ac<0,

Therefore, the discriminant of the quadratic equation ax2+bx+c=0 is Δ=b2-4ac>0.

Therefore, the quadratic equation of one variable must have two unequal real roots. Let them be x1 and x2, then x1x2= <0,

So the two roots of the equation have different signs.

That is, the equation ax2+bx+c=0 has one positive root and one negative root.

necessity:

A quadratic equation of one variable has one positive root and one negative root, let x1, x2,

Then from the relationship between roots and coefficients, x1x2= <0, that is, ac<0,

In summary, it can be seen that 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.

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Update Time: 2024-10-09

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