Lgica yDemostraciones

enMatemtica

Alvaro M. Naupay Gusukuma

Copyright c 2013 Naupay Gusukuma

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Licensed under the Creative Commons Attribution-NonCommercial 3.0 Unported License (theLicense). You may not use this file except in compliance with the License. You may obtain acopy of the License at http://creativecommons.org/licenses/by-nc/3.0. Unless requi-red by applicable law or agreed to in writing, software distributed under the License is distributedon an AS IS BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express orimplied. See the License for the specific language governing permissions and limitations underthe License.

First printing, March 2013

ndice general

1 Los Primeros Ejemplos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1 Paragraphs of Text 5

1.2 Citation 5

1.3 Lists 51.3.1 Numbered List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Bullet Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.3 Descriptions and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 In-text Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Theorems 72.1.1 Several equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Single Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Definitions 7

2.3 Notations 8

2.4 Remarks 8

2.5 Corollaries 8

2.6 Propositions 82.6.1 Several equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.2 Single Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.7 Examples 82.7.1 Equation and Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7.2 Paragraph of Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.8 Exercises 9

2.9 Problems 9

2.10 Vocabulary 9

3 Presenting Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1 Table 11

3.2 Figure 11

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Books 13

Articles 13

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Paragraphs of TextCitationLists

Numbered ListBullet PointsDescriptions and Definitions

1. Los Primeros Ejemplos

1.1 Paragraphs of Text

Figura 1.1: Dibujo hecho enteramente en Inkscape

1.2 CitationThis statement requires citation [book_key ]; this one is more specific [article_key ].

1.3 ListsLists are useful to present information in a concise and/or ordered way1.

1.3.1 Numbered List1. The first item2. The second item3. The third item

1.3.2 Bullet PointsThe first itemThe second itemThe third item

1Footnote example...

6 Captulo 1. Los Primeros Ejemplos

1.3.3 Descriptions and DefinitionsName DescriptionWord DefinitionComment Elaboration

TheoremsSeveral equationsSingle Line

DefinitionsNotationsRemarksCorollariesPropositions

Several equationsSingle Line

ExamplesEquation and TextParagraph of Text

ExercisesProblemsVocabulary

2. In-text Elements

2.1 Theorems

This is an example of theorems.

2.1.1 Several equations

This is a theorem consisting of several equations.

Teorema 2.1.1 Name of the theorem. In E = Rn all norms are equivalent. It has theproperties:||x|| ||y|| ||xy|| (2.1)

||n

i=1

xi|| n

i=1||xi|| where n is a finite integer (2.2)

2.1.2 Single Line

This is a theorem consisting of just one line.

Teorema 2.1.2 A set D(G) in dense in L2(G), | |0.

2.2 Definitions

This is an example of a definition. A definition could be mathematical or it could define aconcept.

Definicin 2.2.1 Definition name. Given a vector space E, a norm on E is an application,denoted || ||, E in R+ = [0,+[ such that:

||x||= 0 x= 0 (2.3)||x||= | | ||x|| (2.4)||x+y|| ||x||+ ||y|| (2.5)

8 Captulo 2. In-text Elements

2.3 NotationsNotacin 2.1. Given an open subset G of Rn, the set of functions are:

1. Bounded support G;2. Infinitely differentiable;

a vector space is denoted by D(G).

2.4 RemarksThis is an example of a remark.

R The concepts presented here are now in conventional employment in mathematics. Vectorspaces are taken over the field K= R, however, established properties are easily extendedto K= C.

2.5 CorollariesThis is an example of a corollary.

Corolario 2.5.1 Corollary name. The concepts presented here are now in conventio-nal employment in mathematics. Vector spaces are taken over the field K = R, however,established properties are easily extended to K= C.

2.6 PropositionsThis is an example of propositions.

2.6.1 Several equationsProposicin 2.6.1 Proposition name. It has the properties:||x|| ||y|| ||xy|| (2.6)

||n

i=1

xi|| n

i=1||xi|| where n is a finite integer (2.7)

2.6.2 Single LineProposicin 2.6.2 Let f ,g L2(G); if D(G), ( f ,)0 = (g,)0 then f = g.

2.7 ExamplesThis is an example of examples.

2.7.1 Equation and Text Ejemplo 2.1 Let G = {x R2 : |x|< 3} and denoted by: x0 = (1,1); consider the function:

f (x) =

{e|x| si |x x0| 1/20 si |x x0|> 1/2 (2.8)

The function f has bounded support, we can take A = {x R2 : |x x0| 1/2+ } for all ]0;5/22[.

2.8 Exercises 9

2.7.2 Paragraph of Text Ejemplo 2.2 Example name.

2.8 ExercisesThis is an example of an exercise.

Ejercicio 2.1 This is a good place to ask a question to test learning progress or further cementideas into students minds.

2.9 ProblemsProblema 2.1 What is the average airspeed velocity of an unladen swallow?

2.10 VocabularyDefine a word to improve a students vocabulary.

Vocabulario 2.1 Word. Definition of word.

TableFigureBooksArticles

3. Presenting Information

3.1 Table

Treatments Response 1 Response 2

Treatment 1 0.0003262 0.562Treatment 2 0.0015681 0.910Treatment 3 0.0009271 0.296

Cuadro 3.1: Table caption

3.2 Figure

Figura 3.1: Figure caption

Bibliography

Books

Articles

ndice alfabtico

C

Citation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

D

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

E

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Equation and Text . . . . . . . . . . . . . . . . . . .8Paragraph of Text . . . . . . . . . . . . . . . . . . . 9

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

F

Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

L

Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Bullet Points . . . . . . . . . . . . . . . . . . . . . . . 5Descriptions and Definitions . . . . . . . . . 6Numbered List . . . . . . . . . . . . . . . . . . . . . 5

N

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

P

Paragraphs of Text . . . . . . . . . . . . . . . . . . . . . . 5Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Several Equations . . . . . . . . . . . . . . . . . . . 8Single Line . . . . . . . . . . . . . . . . . . . . . . . . .8

R

Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

T

Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Several Equations . . . . . . . . . . . . . . . . . . . 7Single Line . . . . . . . . . . . . . . . . . . . . . . . . . 7

V

Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Los Primeros EjemplosParagraphs of TextCitationListsNumbered ListBullet PointsDescriptions and Definitions

In-text ElementsTheoremsSeveral equationsSingle Line

DefinitionsNotationsRemarksCorollariesPropositionsSeveral equationsSingle Line

ExamplesEquation and TextParagraph of Text

ExercisesProblemsVocabulary

Presenting InformationTableFigure

BibliographyBooksArticles

Index