lesson 1

42問 • 1年前

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通報

問題一覧

One of the foremost contribution of ancient ___ mathematicians to the foundations of mathematics was the ____

greek axiomatic method & notion of proof

___ deals with the measurement and computation of lengths of line segments, areas of figures, and sizes of angles.

geometry

The word geometry comes from the word ___ which means ___ earth and ____ which means

geo means earth metros to measure

____is originally land surveying.

geometry

Original name of geometry

land surveying

Father of Geometry

euclid

Euclid, famously called the Father of Geometry, lived in the city of ____ around ___ (date)

Alexandria, Egypt 300 BC

He invented the form of mathematical proof that is still used today.

euclid

Name of book. Euclid gathered up all of the known mathematics of his time, as well as a lot of his own, and then he subjected it all to logical, mathematic proofs.

"Elements,"

Euclid's 5 Geometric Postulates:

1. It is possible to draw a straight line from any point to any point. 2. It is possible to extend a finite straight line continuously in a straight line. 3. It is possible to create a circle with any center and distance. 4. All right angles are equal to one another. 5. If a straight line falls on two straight lines, making the interior angles on the same side less than two right angle: the straight lines, if produced indefinitely, will meet on that side where the angles are less than two right angles

is a basic statement assumed to be true and requiring no proof of its truthfulness.

axion

are statements that are considered true without proof or validation.

Postulates (in Geometry)

are statements proved to be true using postulates, definitions, other established theorems, and logic.

theorems

is a list of undefined terms together with a list of statements called axioms that are presupposed to be true.

axiomatic system

is any statement that can be proven using logical deduction from the axioms.

theorem

An axiomatic system consists of:

1. undefined terms 2. axioms/postulates 3. defined terms 4. theorems

concepts accepted without definition/explanation.

undefined terms

In the setting of basic geometry these migh include line or point.

undefined terms

forms the basis of the necessary technical vocabulary

undefined terms

logical statements regarding the undefined terms which are accepted without proof.

axioms/postulate

A set of unproved initial assumptions.

axioms/postulate

For example: There exists a line joining any two given points.

axioms/postulate

concepts defined in terms of 1 and 2.

defined terms

For instance a triangle could be defined in terms of three non-collinear points.

defined terms

logical statements deduced from 1-3.

theorems

expressing properties of the undefined objects, which are derived from the axioms by the laws of logic

theorem

Find the undefined terms: Axiom 1: Every robot has at least two paths Axiom 2: Every path has at least two robots Axiom 3: A minimum of one robot exists

robot and path

Find the undefined terms: Axiom 1: Each committee is a set of three members. AxIom Z: cach member is on exactiv two committees AXIOM 3: NO two members may be together on more than one committee. Axiom 4: There is at least one committee.

member & committee

is an argument demonstrating the truth of a theorem within an axiomatic system.

proof

is an interpretation of the undefined terms such that all the axioms/postulates are true.

model

Properties of Axiomatic Systems

1. consistency 2. independence 3. completeness

has internal logic that is not self-contradictory

consistency

If there is a model for an axiomatic system, then the system is called___

consistent

if it cannot be proven from the other axioms.

independence

if every true statement can be proven from the axioms.

complete

___set out a list of 23 unsolved mathematical problems to focus the direction of research in the 20th Century.

david hilbert

——developed a more modern axiomatic system, and his key idea was to detach "points", "lines" and "planes" from real-world objects.

david hilbert

who reduced geometry to a series of axioms and contributed substantially to the establishment of the formalistic foundations of mathematics.

david hilbert

Basic terms of hilbert:

1. point 2. line 3. plane

Basic relations of hilbert:

1. betweenness 2. a ternary relation linking points lies on (containment) 3. congruence (denoted by an infix #)

Hilbert grouped his axioms into the following groups and we will follow his approach:

1. connection 2. order 3. congruence 4. parallels 5. continuity

As mentioned above, David Hilbert (1862 - 1943) proposed an axiomatic system dealing with geometrical objects like "points", "straight lines", and "planes", detaching them from their specific meaning. Hilbert wanted to avoid any attempt to define exactly what a point, a straight line, or a plane is. He rather focused on the relationships between these objects. The following definitions and axioms demonstrate this approach.

Axioms of Connection and Their Consequences

MATH finals Module 8 Quantifiers and Negations

ユーザ名非公開 · 7問 · 2年前

MATH finals Module 8 Quantifiers and Negations

7問 • 2年前

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STS prelim module 1 (ancient) part 1

ユーザ名非公開 · 100問 · 2年前

STS prelim module 1 (ancient) part 1

100問 • 2年前

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STS prelim module 1 (ancient) part 2

ユーザ名非公開 · 26問 · 2年前

STS prelim module 1 (ancient) part 2

26問 • 2年前

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STS prelim module 1 (medieval)

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STS prelim module 1 (medieval)

6問 • 2年前

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STS prelim module 1 (modern)

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STS prelim module 1 (modern)

STS prelim phil

STS prelim phil

Contemp Module 1 Prelim

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Contemp Module 1 Prelim

Lesson 1 Part 1

Lesson 1 Part 1

Lesson 1 Part 2

Lesson 1 Part 2

lesson2

lesson2

environmental factors

ユーザ名非公開 · 5問 · 2年前

environmental factors

5問 • 2年前

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organizational factor

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organizational factor

M1 Lesson 1

M1 Lesson 1

M1 Lesson 2 Functions of Art

ユーザ名非公開 · 31問 · 2年前

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Yunit 2

Yunit 2

retorika intro

retorika intro

Module 1

Module 1

Ancient World

Ancient World

medieval/middle age

medieval/middle age

modern ages

modern ages

into

into

globalization 1st part

ユーザ名非公開 · 24問 · 2年前

globalization 1st part

globalization pt2

globalization pt2

globalization 2nd part

ユーザ名非公開 · 35問 · 1年前

globalization 2nd part

Philippines

Philippines

Philippines inventors

ユーザ名非公開 · 7問 · 1年前

Philippines inventors

Yunit 3

Yunit 3

Yunit 3 part 2

Yunit 3 part 2

structures of globalization

ユーザ名非公開 · 93問 · 1年前

structures of globalization

93問 • 1年前

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SCIENCETECHNOLOGY ANDNATIONBUILDING

ユーザ名非公開 · 64問 · 1年前

SCIENCETECHNOLOGY ANDNATIONBUILDING

64問 • 1年前

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Filipino inventions (video)

ユーザ名非公開 · 15問 · 1年前

Filipino inventions (video)

15問 • 1年前

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Government Policies on Science and Technology

ユーザ名非公開 · 31問 · 1年前

Government Policies on Science and Technology

match match

match match

Yunit 1

Yunit 1

Yunit 1 (quote)

Yunit 1 (quote)

lesson 3

lesson 3

lesson 5

lesson 5

lesson 4

lesson 4

Chapter 1 part 1

Chapter 1 part 1

Chapter 1 part 2

Chapter 1 part 2

chapter 1 tanong tanong

ユーザ名非公開 · 11問 · 1年前

chapter 1 tanong tanong

11問 • 1年前

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United Nations Organization (UNO)

ユーザ名非公開 · 76問 · 1年前

United Nations Organization (UNO)

mga mahirap

mga mahirap

physical fitness and wellness

ユーザ名非公開 · 40問 · 1年前

physical fitness and wellness

final ppt

final ppt

fundamental body movements

ユーザ名非公開 · 43問 · 1年前

fundamental body movements

axis of motion

axis of motion

north and south divide

ユーザ名非公開 · 25問 · 1年前

north and south divide

asian regionalism

asian regionalism

GLOBALIZATION OF RELIGION

ユーザ名非公開 · 11問 · 1年前

GLOBALIZATION OF RELIGION

Global Media Culture

Global Media Culture

asian regionalism (book)

ユーザ名非公開 · 22問 · 1年前

asian regionalism (book)

22問 • 1年前

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Global Culture and Media (book)

ユーザ名非公開 · 14問 · 1年前

Global Culture and Media (book)

14問 • 1年前

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lesson 1 ang retorika at diskurso & lesson 2

ユーザ名非公開 · 50問 · 1年前

lesson 1 ang retorika at diskurso & lesson 2

50問 • 1年前

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lesson 3 ORGANISASYON NG DISKURSONGPASALITA AT PASULAT

ユーザ名非公開 · 12問 · 1年前

lesson 3 ORGANISASYON NG DISKURSONGPASALITA AT PASULAT

speech

speech

Intellectual Revolutions that Defined Society

ユーザ名非公開 · 41問 · 1年前

Intellectual Revolutions that Defined Society

41問 • 1年前

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Technology as a Way of Revealing

ユーザ名非公開 · 23問 · 1年前

Technology as a Way of Revealing

Human Flourishing

Human Flourishing

lahat na

lahat na

the good life

the good life

When Technology and Humanity Cross

ユーザ名非公開 · 45問 · 1年前

When Technology and Humanity Cross

mga kulang

mga kulang

Lesson 1: Information and Communication Technology

ユーザ名非公開 · 9問 · 1年前

Lesson 1: Information and Communication Technology

tayutay kinds

tayutay kinds

tayutay general

tayutay general

tayutay kinds p2

tayutay kinds p2

history of computer 11-20/20

ユーザ名非公開 · 27問 · 1年前

history of computer 11-20/20

27問 • 1年前

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Basic Computing Periods - Ages

ユーザ名非公開 · 29問 · 1年前

Basic Computing Periods - Ages

generation

generation

MODULE 3: THE WEB AND THE INTERNET

ユーザ名非公開 · 39問 · 1年前

MODULE 3: THE WEB AND THE INTERNET

types of websites

types of websites

lesson 2: the internet

ユーザ名非公開 · 44問 · 1年前

lesson 2: the internet

44問 • 1年前

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history of computer 1-10/20

ユーザ名非公開 · 33問 · 1年前

history of computer 1-10/20

lesson 1 books

lesson 1 books

FLEXIBILITY TRAININGEXERCISES

ユーザ名非公開 · 54問 · 1年前

FLEXIBILITY TRAININGEXERCISES

pilates

pilates

elements of literature

ユーザ名非公開 · 96問 · 1年前

elements of literature

elements of poetry

elements of poetry

elements of short story

ユーザ名非公開 · 26問 · 1年前

elements of short story

elements of novel

elements of novel

elements of essay

elements of essay

elements of literature (2)

ユーザ名非公開 · 83問 · 1年前

elements of literature (2)

general

general

elements of drama

elements of drama

sculpture

sculpture

tao

tao

3 gec 8 finals (PPT)

3 gec 8 finals (PPT)

module 8 why the future doesn’t need us

ユーザ名非公開 · 23問 · 1年前

module 8 why the future doesn’t need us

23問 • 1年前

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module 9 The Information Age (Gutenberg to Social Media)

ユーザ名非公開 · 59問 · 1年前

module 9 The Information Age (Gutenberg to Social Media)

59問 • 1年前

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MODULE 10Biodiversity and Healthy Society

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MODULE 10Biodiversity and Healthy Society

32問 • 1年前

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MODULE 11 Genetically Modified Organisms: Science, Health and Politics

ユーザ名非公開 · 40問 · 1年前

MODULE 11 Genetically Modified Organisms: Science, Health and Politics

40問 • 1年前

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module 11 continuation

ユーザ名非公開 · 19問 · 1年前

module 11 continuation

19問 • 1年前

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問題一覧

One of the foremost contribution of ancient ___ mathematicians to the foundations of mathematics was the ____

greek axiomatic method & notion of proof

___ deals with the measurement and computation of lengths of line segments, areas of figures, and sizes of angles.

geometry

The word geometry comes from the word ___ which means ___ earth and ____ which means

geo means earth metros to measure

____is originally land surveying.

geometry

Original name of geometry

land surveying

Father of Geometry

euclid

Euclid, famously called the Father of Geometry, lived in the city of ____ around ___ (date)

Alexandria, Egypt 300 BC

He invented the form of mathematical proof that is still used today.

euclid

Name of book. Euclid gathered up all of the known mathematics of his time, as well as a lot of his own, and then he subjected it all to logical, mathematic proofs.

"Elements,"

Euclid's 5 Geometric Postulates:

is a basic statement assumed to be true and requiring no proof of its truthfulness.

axion

are statements that are considered true without proof or validation.

Postulates (in Geometry)

are statements proved to be true using postulates, definitions, other established theorems, and logic.

theorems

is a list of undefined terms together with a list of statements called axioms that are presupposed to be true.

axiomatic system

is any statement that can be proven using logical deduction from the axioms.

theorem

An axiomatic system consists of:

1. undefined terms 2. axioms/postulates 3. defined terms 4. theorems

concepts accepted without definition/explanation.

undefined terms

In the setting of basic geometry these migh include line or point.

undefined terms

forms the basis of the necessary technical vocabulary

undefined terms

logical statements regarding the undefined terms which are accepted without proof.

axioms/postulate

A set of unproved initial assumptions.

axioms/postulate

For example: There exists a line joining any two given points.

axioms/postulate

concepts defined in terms of 1 and 2.

defined terms

For instance a triangle could be defined in terms of three non-collinear points.

defined terms

logical statements deduced from 1-3.

theorems

expressing properties of the undefined objects, which are derived from the axioms by the laws of logic

theorem

Find the undefined terms: Axiom 1: Every robot has at least two paths Axiom 2: Every path has at least two robots Axiom 3: A minimum of one robot exists

robot and path

member & committee

is an argument demonstrating the truth of a theorem within an axiomatic system.

proof

is an interpretation of the undefined terms such that all the axioms/postulates are true.

model

Properties of Axiomatic Systems

1. consistency 2. independence 3. completeness

has internal logic that is not self-contradictory

consistency

If there is a model for an axiomatic system, then the system is called___

consistent

if it cannot be proven from the other axioms.

independence

if every true statement can be proven from the axioms.

complete

___set out a list of 23 unsolved mathematical problems to focus the direction of research in the 20th Century.

david hilbert

——developed a more modern axiomatic system, and his key idea was to detach "points", "lines" and "planes" from real-world objects.

david hilbert

who reduced geometry to a series of axioms and contributed substantially to the establishment of the formalistic foundations of mathematics.

david hilbert

Basic terms of hilbert:

1. point 2. line 3. plane

Basic relations of hilbert:

1. betweenness 2. a ternary relation linking points lies on (containment) 3. congruence (denoted by an infix #)

Hilbert grouped his axioms into the following groups and we will follow his approach:

1. connection 2. order 3. congruence 4. parallels 5. continuity

Axioms of Connection and Their Consequences