"Universal Quantifiers and Existential Quantifiers" Sets and Commonly Used Logical Terms PPT Courseware

Description

"Universal Quantifiers and Existential Quantifiers" Sets and Commonly Used Logical Terms PPT Courseware

Part One: Learning Objectives

Understand the definitions of universal quantifiers and universal quantifier propositions, and understand the definitions of existential quantifiers and existential quantifier propositions.

Master the method of judging whether universal quantifier propositions and existential quantifier propositions are true or false

Understand the relationship between universal quantifier propositions and existential quantifier propositions, and master the method of negating universal quantifier propositions or existential quantifier propositions

Universal quantifiers and existential quantifiers PPT, part 2: independent learning

Problem guide

Preview textbooks P24-P29 and think about the following questions:

1. What are the definitions of universal quantifier and universal quantifier proposition?

2. What are the definitions of existential quantifiers and existential quantifier propositions?

3. What are the negations of universal quantifier propositions and existential quantifier propositions?

4. What is the negation of the universal quantifier proposition "∀x∈M, p(x)"?

5. What is the negation of the existential quantifier proposition "∃x∈M, p(x)"?

A preliminary exploration of new knowledge

1. Universal quantifiers and existential quantifiers

■Instructions from famous teachers

(1) A universal quantifier proposition is a proposition that states that all elements in a certain set have certain properties. Common universal quantifiers include "everything", "every one", "any give", etc.

(2) The existential quantifier proposition is a proposition that states that there is one or some elements in a certain set with certain properties. Common existential quantifiers include "some", "a certain one", "some", etc.

2. Negation of universal quantifier propositions and existential quantifier propositions

■Instructions from famous teachers

(1) To deny the universal quantifier proposition "∀x∈M, p(x)", you only need to find an x ​​in M ​​such that p(x) does not hold, that is, the proposition "∃x∈M, �p(x) "Established.

(2) To deny the existence of the quantifier proposition "∃x∈M, p(x)", it is necessary to verify that for every x in M, p(x) is not true, that is, the proposition "∀x∈M, �p( x)" is established.

[ Reminder p(x), at the same time, change the properties of the quantifier, that is, change the universal quantifier to an existential quantifier, and the existential quantifier to a universal quantifier.

self prediction

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

(1) Phrases such as "some", "some" and "some" are not existential quantifiers. ()

(2) The universal quantifier means "arbitrariness", and the existential quantifier means "existence". ()

(3) Universal quantifier propositions must contain universal quantifiers, and existential quantifier propositions must contain existential quantifiers. ()

(4) The truth and falsity of ∃x∈M, p(x) and ∀x∈M, �p(x) are opposite. ()

The following sentences have quantifier propositions ()

A． The integer n is a multiple of 2 and 5

B． There exists an integer n such that n is divisible by 11

C. If 3x-7=0, then x=73

D. ∀x∈M,p(x)

The negation of the proposition "For any x∈R, x3-x2+1≤0" is ()

A． There is no x∈R, x3-x2+1≤0

B． There exists x∈R, x3-x2+1≥0

C. For any x∈R, x3-x2+1>0

D. There exists x∈R, x3-x2+1>0

Universal quantifiers and existential quantifiers PPT, the third part: lecture and practice interaction

Discrimination between universal quantifier propositions and existential quantifier propositions

Determine whether the following sentences are universal quantifier propositions or existential quantifier propositions.

(1) The solution set A of all inequalities satisfies A⊆R;

(2) Some real numbers a and b can make |a-b|=|a|＋|b|;

(3) For any a, b∈R, if a>b, then 1a<1b;

(4) The square of a natural number is a positive number.

regular method

Determine whether a statement is a universal quantifier proposition

Or is there an idea of ​​quantifier proposition?

[Note] The universal quantifier can be omitted for universal quantifier propositions, but the existential quantifier for existential quantifier propositions generally cannot be omitted.

Track training

1. Give the following proposition:

①There is a real number x>1, so that x2>1;

②Congruent triangles must be similar;

③Some similar triangles are congruent;

④ There is at least one real number a such that the root of ax2-ax+1=0 is a negative number.

The number of quantifier propositions among them is ()

A． 1B． 2

C. 3D． 4

2. Use the quantifier symbols "∀" and "∃" to express the following propositions.

(1) All real numbers x can establish x2+x+1>0;

(2) For all real numbers a, b, the equation ax + b = 0 has exactly one solution;

(3) There must be integers x and y such that 3x-2y=10 is true;

(4) All rational numbers x can make 13x2+12x+1 a rational number.

True or False Judgment of Universal Quantifier Propositions and Existential Quantifier Propositions

Decide whether the following propositions are true or false.

(1)∃x∈Z, x3<1;

(2) There is a quadrilateral that is not a parallelogram;

(3) In the plane rectangular coordinate system, any ordered pair of real numbers (x, y) corresponds to a point P;

(4)∀x∈N, x2>0.

regular method

Methods to determine whether universal quantifier propositions and existential quantifier propositions are true or false

(1) To judge that a universal quantifier proposition is true, the proposition p(x) must be true for every element x in the given set; but to judge that a universal quantifier proposition is false, as long as the given set Find an element x in the set that makes the proposition p(x) false.

(2) To judge that an existential quantifier proposition is true, you only need to find an element x in the given set to make the proposition p(x) true; to judge that an existential quantifier proposition is false, you must find an element x in the given set Every element x makes the proposition p(x) false.

Universal quantifiers and existential quantifiers PPT, Part 4: Feedback on compliance with standards

1. Which of the following four propositions is both an existential quantifier proposition and a true proposition ()

A． Are the interior angles of an acute triangle acute or obtuse?

B． There is at least one real number x such that x2≤0

C. The sum of two irrational numbers must be an irrational number

D. There exists a negative number x such that 1x>2

2. The following proposition is another way of expressing "∀x∈R, x2>3" ()

A． There is an x∈R such that x2>3

B． For some x∈R, such that x2>3

C. Choose any x∈R such that x2>3

D. There is at least one x∈R such that x2>3

3. The negation of the proposition "For any x∈R, x3-x2+2<0" is ()

A． There is no x∈R, x3-x2+2≥0

B． There exists x∉R, x3-x2+2≥0

C. There exists x∈R, x3-x2+2≥0

D. There exists x∈R, x3-x2+2<0

4. Decide whether the following propositions are true or false.

(1) The length of each line segment can be expressed by a positive rational number;

(2) There exists a real number x such that the equation x2+x+8=0 holds.

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Update Time: 2024-11-18

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