OpenLLM-France/wikimedia huggingface dataset:
# Canadian Refugee Procedure/IRPR s. 13 - Documents and Certified Copies
## IRPR s. 13

The text of the relevant section of the regulation reads:
# Algebra/Chapter 7

Chapter 7: Systems and Matrices
← Chapter 6 · Chapter 8 →
Introduction

## Sections

* Systems of Equations
* Systems of Inequalities
* Systems with Three Variables
* Matrices
* Matrix Algebra
* Determinants
* Inveres of Matrices
* Solving Systems with Matrices
* Linear Programming

## Extra

* Review
* Practice Problems
* Solutions to Practice Problems
* Glossary
# Algebra/Chapter 7/Exercises

A set of exercises related to concepts from Chapter 7.
This set contains 5 problems (0 Conceptual Questions + 5 Exercises + 0 Projects)

## Exercises
### Section 7.1

(★) 7.1 (A Sum and a Difference) The sum of two numbers is 104 and their difference is 32. What is the value of the larger number?
(★) 7.2 (Solving Systems of Equations by Graphing) Solve each of the systems of equations by graphing.
(★) 7.3 (Solving Systems of Equations by Substitution) Solve each of the systems of equations by substitution.
(★) 7.4 (Solving Systems of Equations by Elimination) Solve each of the systems of equations by elimination.
(★) 7.5 (a + b) Find the value of a + b if 3a + 7b = 1977 and 5a + b = 2007.
# Algebra/Chapter 10

Chapter 10: Polynomial Functions
← Chapter 9 · Chapter 11 →
Introduction

## Sections

* Polynomials
* Adding and Subtracting Polynomials
* Multiplying Polynomials
* The Binomial Theorem
* Graphing Polynomials
* The Fundamental Theorem of Algebra
* Polynomial Inequalities

## Extra

* Review
* Practice Problems
* Solutions to Practice Problems
* Glossary
# Arithmetic/Cover

Chapter 0 →

# Welcome To Arithmetic!

This wikibook will teach you arithmetic and pre-algebra. If you wish to read this book, click the link above.
# Czech/Alphabet and Pronunciation

Czech alphabet consists of 42 normal Latin letters, some have an accent:
* All vowels can be either short (aeiouy), or long with acute (čárka) (áéěíóúůý). (This can denote a different word.)
* The only difference between Ú and Ů is that Ú is used only at the beginning of a word (or a part of a word like: triangle = trojúhelník), while Ů can only be in the middle of a word.
* There are also "softened" versions of the letters D E N R S T Z, with a hook (háček): ˇ
* The accents lengthen the vowels, but they do not imply stress. Stress is almost always on the first syllable.
A Á B C Č D Ď E É Ě F G H Ch I Í J K L M N Ň O Ó P Q R Ř S Š T Ť U Ú Ů V W X Y Ý Z Ž
Note: Except for foreign words, the letters Q, W and X are almost never used, and the first two are often replaced with Kv and V. F, G and Ó also mostly occur in words of foreign origin, but are nowhere near as rare as Q, W, or X.
Most of the letters are pronounced similarly to in English. Note:
* Ch is pronounced as "kh" and considered as one letter
* C like in the 'zz' in pizza or German 'z' as in 'Zimmer'
* Ď like /ɟ/ (D, but with your tongue touching the entire roof of your mouth) or Polish 'gi'
* E and G like in "beggar"
* H like in "head"
* J like the 'y' in "yeah"
* Ň like in "menu" or the Spanish 'señor'
* R somewhat harder than in English, a bit like in Spanish "arriba" initially and rolled in the middle of a word.
* Ť like /c/ (T, but with your tongue touching the entire roof of your mouth) or Polish 'ki'
The pronunciation rarely changes depending on the position, except for:
* D, N, T are pronounced as Ď, Ň, Ť before I, Í, or Ě Note: This is the only reason why Ě is used. The letter itself is pronounced as E. Not quite true, consider following example: "válka měla mnoho obětí" and "těšila se domů na jeho objetí..." etc.
* Czech has word-final devoicing of consonants. This means that in speech, voiced consonants are pronounced as their voiceless counterparts: D → T, Ď → Ť, H → Ch, G → K, V → F, B → P, Z → S, Ž → Š.
Other notes:
Now you know almost everything. You can go to the Czech pronunciation page and hear it.
# Administrative divisions of Mongolia

Mongolia is a country with extensive landscape which is populated mainly by Mongolian-speaking people. Other Mongolic-speaking people live in Inner Mongolia and other scattered lands in China plus Buryatia and Kalmykia in Russia. After a certain time playing a major role in world politics starting from the 1206's establishment of the Mongol Empire which conquered most of Eurasia, Mongol identity and its achievements became less popular until the only Mongolic-speaking people left were invaded by Manchu Qing dynasty by mid- and late-1600s. Course of events following the fall of the Qing dynasty, resulted in Mongol people's self-governance of modern Mongolia by the beginning of 20-th century. Throughout the 20-th century, Mongolia had a diverse path in the history as well as shaping of its administrative divisions.

## Contents

* Part 1: Vertical Axis
** Chapter 1: Ulaanbaatar
** Chapter 2: North: Selenge, Darkhan, and Tuv
** Chapter 3: Northwest: Bulgan, Erdenet and Huvsgul
** Chapter 4: South: The vast Gobi
* Part 2: Eastern region
** Chapter 5: Khentii
** Chapter 6: Sukhbaatar
** Chapter 7: Dornod
* Part 3: Khangai region
** Chapter 8: Arkhangai, Uvurkhangai and Bayankhongor
* Part 4: Taishir region
** Chapter 9: Zavkhan
** Chapter 10: Govi-Altai
* Part 5: Western region
** Chapter 11: Khovd
** Chapter 12: Uvs
** Chapter 13: Bayan-Ulgii
# Russian/Lesson 2

Russian language · Русский язык :
| Lessons | Introduction · Alphabet · Lesson 1 · Lesson 2 · Lesson 3 · Lesson 4 · Lesson 5 | (view) |
| Reference | Numbers · Cases (Nom. · Gen. · Dat. · Acc. · Inst. · Prep.) · Adjectives · Prepositions · Verbs (Aspect · Past · Future) · Pronouns (Personal · Possessive · Interrogative) · Cursive | |
| Appendices | Appendix · Alphabet · Internet · Cheat Sheet | |

## Произношéние - Pronunciation

Usually, Russian is pronounced as you see it. However, there are some exceptions, usually changes in the pronunciation of vowels based on where they are in relation to the stressed syllable.
    O sounds like the 'o' in 'rogue' (i.e., 'oh') when it is stressed. An o that is just before the stress is pronounced like the 'a' in far (i.e., 'ah'). Otherwise, it's pronounced like the 'a' in 'about' (i.e., 'uh'). So, the word хорошо́ is pronounced: /xərɐˈʂo/.
    E is pronounced 'ye', as in 'yes', when stressed. When it's not stressed, e sounds like и, as in the 'e' in 'piglet'.
If you pronounce these letters without reducing them you will be understandable, but will sound strange.
Voiced consonants at the end of the word become unvoiced. So, таз sounds like 'tas', взвод sounds like 'vzvot', etc. The same thing happens if a voiced consonant is followed by an unvoiced one. For example, подско́к sounds like 'patskók', and водка sounds like 'votka'.

## Диалóг - Dialogue

Unlike modern English, Russian has two words for 'you': ты and вы. You use вы when the person you're talking to is someone you don't know, someone you want to show respect to (such as a teacher or elder), or a group of people. In this sense, it is both a more polite form of ты, and a plural form. So you would only use ты to talk to a single person who you know well. In written Russian, a capitalised Вы is used when being respectful, while the lower-case вы is used when talking to many people. You can also listen (help·info) to the audio version of this dialogue.

### Здра́вствуйте, я рад/ра́да Вас ви́деть!

    Zdrastvuyti, ya rat/rada Vas vidjet
    Hello, I am glad to see you!
Note: The Russian for 'glad' is рад ('rat') when spoken by a male, or ра́да (rada) when spoken by a female

### До́брый день, и я то́же.

    Dobry denj, i ya tozhi
    Good afternoon, me too.

### Меня́ зовут́ Джо́ан, а Вас?

    Minya zavut Joanne, ah Vas?
    My name is Joanne, and yours?
Note: This literally means "Me (they) call Joanne, and you?", or "They call me Joanne, and you?" and is the Russian construction of asking someone's name.

### О́чень прия́тно!

    Ochin' priyatna
    Pleased to meet you. (Literally, "[it is] very pleasant").

### А меня́ — Ма́рья Степа́новна, мо́жно про́сто Ма́ша.

    A minya – Maryah Stipanavna, mozhna prosto Masha
    And mine is Marya Stepanovna, one may call me just Masha. (Literally, "and [they call] me Marya Stepanovna, [one] may [call me] just Masha").

### Спаси́бо, до встре́чи!

    Spasiba, da fstrechi
    Thank you, see you again! (Literally, "thank you, till another meeting").

### До свида́ния.

    Da svidanya
    Good bye. (Literally, "till another seeing").
Lesson 3 >>
# Algebra/Chapter 24

Chapter 24: Limits of Functions
← Chapter 23 · Chapter 25 →
Introduction

## Sections

* Limits
* One-Sided Limits
* Limit Properties
* Limits at Infinity
* Limits of Sequences
* Continuity
* The Definition of a Limit
* Derivatives
* Anti-Derivatives and Integrals

## Extra

* Review
* Practice Problems
* Solutions to Practice Problems
* Glossary
# Algebra/Chapter 25

Chapter 25: Group Theory
← Chapter 24 · Chapter 26 →
Introduction

## Sections

* Symmetries
* Binary Operators
* Groups
* Lagrange's Theorem
* Ring Theory

## Extra

* Review
* Practice Problems
* Solutions to Practice Problems
* Glossary
# Algebra/Chapter 26

Chapter 26: Graphs and Networks
← Chapter 25 · Chapter 27 →
Introduction

## Sections

* Graphs and Networks
* Euler Paths
* Hamiltonian Paths
* Chromatic Numbers
* Planar Graphs
* TSP Graphs
* The Shortest Route Algorithm
* Tree Graphs
* Minimum Spanning Trees

## Extra

* Review
* Practice Problems
* Solutions to Practice Problems
* Glossary
# Algebra/Chapter 23

Chapter 23: Higher-Degree Polynomials
← Chapter 22 · Chapter 24 →
Introduction

## Sections

* Polynomial Roots
* Cubic Equations
* Quartic Equations
* Quintic Equations
* Sextic and Septic Equations
* Properties of Polynomial Roots
* Higher-Degree Curves
* Fermat's Last Theorem

## Extra

* Review
* Practice Problems
* Solutions to Practice Problems
* Glossary
# Yoruba/Numbers

1- Oókan 2- Eéjì 3 - Ẹẹ́ta 4 - Ẹẹ́rin 5- Aárùn-ún 6- Ẹẹ́fà 7- Eéje 8 - Ẹẹ́jọ 9 - Ẹẹ́sàán 10- Ẹẹ́wàá 11 - Ọọ́kànlá 12- Eéjìlá 13- Ẹẹ́tàlá 14 - Ẹ́rìnlá 15 - Márùndínlógún/Mẹ́ẹ̀ẹ́dógún 16 - Ẹ́rìndínlógún 17 - Ẹ́tàdínlógún 18 - Éjìdínlógún 19 - Ọ́kàndínlógún 20 - Ogún 21 - Ọ́kànlélógún 22 - Méjìlélógún 23 - Mẹ́tàlélógún 24 - Mẹ́rìnlélógún 25 - Márùndínlọ́gbọ̀n 26 - Mẹ́rìndínlọ́gbọ̀n 27 - Mẹ́tàdínlọ́gbọ̀n 28 - Méjìdínlọ́gbọ̀n 29 - Ọ̀kándínlọ́gbọ̀n 30 - Ọgbọ̀n 40 - Ogójì 50 - Àádọ́ta 60 - Ọgọ́ta 70 - Àádọ́rin 80 - Ọgọ́rin 90 - Àádọ́rùn-ún 100 - Ọgọ́rùn-ún 101 - Ọ́kàn lé lọ́gọ́rùn-ún 102 - Méjì lé lọ́gọ́rùn-ún 103 - Mẹ́tà lé lọ́gọ́rùn-ún
104 - Mẹ́rìn lé lọ́gọ́rùn-ún
105 - Márùn dín láàádọ́fà 110 - Àádọ́fà 120 - Ọgọ́fà 130 - Àádọ́je 140 - Ọgọ́je 150 - Àádọ́jọ 160 - Ọgọ́jọ 170 - Àádọ́sàn-án 180 - Ọgọ́sàn-án 190 - Àádọ́wàá 200 - Igba 300 - Ọ̀ọ́dúnrún 400 - Irinwó 500 - Ẹ̀ẹ́dẹ́gbẹ̀ta 600 - Ẹgbẹ̀ta 700 - Ẹ̀ẹ́dẹ́gbẹ̀rin 800 - Ẹgbẹ̀rin 900 - Ẹ̀ẹ́dẹ́gbẹ̀rún 1000 - Ẹgbẹ̀rún 2000 - Ẹgbẹ̀wá/Ẹgbàá 2500 - Ẹ̀ẹ́dẹ́gbẹ̀tàlá 3000 - Ẹgbẹ̀ẹ̀ẹ́dógún 4000 - Ẹgbààjì 5000 - Ẹ̀ẹ́dẹ́gbàta 6000 - Ẹgbàáta 7000 - Ẹ̀ẹ́dẹ́gbàrin 8000 - Ẹgbàárin 9000 - Ẹ̀ẹ́dẹ́gbàrún 10000 - Ẹgbààrún 20000 - Ọ̀kẹ́ 50000 - Ọ̀kẹ́ méjì (àti) ààbọ̀ 100000 - Ọ̀kẹ́ márùn-ún 1000000 - Àádọ́ta ọ̀kẹ́
For more information on Yorùbá numbers, visit Wikipedia's page on Yorùbá Numerals or the website Yoruba Numeral.
# Basic Algebra/Introduction to Basic Algebra Ideas/Working With Negative Numbers
## Lesson
### Negative Numbers

A positive number is a number greater than zero.
A negative number is a number less than zero. You make a negative number by doing the negative operation on a positive number. You use the " – " sign for the negative operation. This sign is the same you use for subtracting.

### Adding and Subtracting

Adding a negative number is the same as subtracting a positive number.
* $7+(-4)=7-4$
* $x+(-y)=x-y$
Subtracting a negative number is the same as adding a positive number.
* $7-(-4)=7+4$
* $x-(-y)=x+y$

### Multiplying and Dividing

Multiplying a negative number by a positive number, or a positive number by a negative number makes the result negative.
* $(-2)\times 3=-6$
* $2\times (-3)=-6$
Multiplying a negative number by a negative number makes the result positive.
* $(-2)\times (-3)=6$
You do the same for dividing.
* $(-6)\div 3=-2$
* $6\div (-3)=-2$
* $(-6)\div (-3)=2$

### Exponentiating

Exponentiating a negative number to an even (a number you can divide by two) power makes the result positive.
* $(-3)^{2}=9$
* $(-x)^{2}=(-x)\times (-x)=x^{2}$
Exponentiating a negative number to an odd(a number you can not divide by two) power makes the result negative.
* $(-2)^{3}=-8$
* $(-x)^{3}=(-x)\times (-x)\times (-x)=x^{2}\times (-x)=-x^{3}$

### Order of Operations

The negative operation has the same precedence as multiplying and dividing.
* $3+8\div 4=3+2=5$
* $-3^{2}=-(3\times 3)=-9$
* $(-3)^{2}=(-3)\times (-3)=9$

## Example Problems

* $4+(-4)=0$
* $4+(-7)=-3$
* $0+(-2)=-2$
* $-5+7-2\times (-4)=10$
# Hebrew/Colors

Note: This page does not currently fit into of the sequences in the Hebrew Wikibook. It can be used as a reference.
# Algebra/Chapter 26/Exercises

A set of exercises related to concepts from Chapter 26.
This set contains 1 exercise (including the Conceptual Questions)

## Exercises
### Section 26.1

26.1 (Counting Edges and Vertices) How many vertices and edges does the graph below have?
# Algebra/Chapter 25/Exercises
## Exercises
### Section 25.1
### Section 25.2

25.1 Assume that * is a binary operation on the set S. Prove the following statements.

### Section 25.3

25.2 Let $G$ be a group. Prove that the identity element $e\in G$ is unique. Also prove that every element $x\in G$ has a unique inverse, indicated by $x^{-1}$.
# Algebra/Chapter 24/Exercises

A set of exercises related to concepts from Chapter 24.
This set contains 3 exercises (including the Conceptual Questions)

## Exercises
### Section 24.1

24.1 (Limit of a Function I) Use the function f(x) to answer the following questions.
24.2 (Limit of a Function II) Use the function g(x) to answer the following questions.
24.3 (Limit of a Function III) Use the function h(x) to answer the following questions.
# Quantum Chemistry/Example 3

Calculate the number of nodes for a particle in a 1D box when n=2 and n=3 when the length of the box is L=5, and give the x-intercept of the node(s).
    Wavefunction for a particle in a 1D-Box $\psi (x)={\sqrt {\frac {2}{L}}}\sin \left({\frac {n\pi x}{L}}\right)$

## Answer
#### Part 1

n$_{nodes}$=n-1
n$_{nodes}$=2-1=1
We're looking for one x-intercept.
$0\div {\sqrt {\frac {2}{5}}}={\sqrt {\frac {2}{5}}}\sin \left({\frac {2\pi x}{5}}\right)\div {\sqrt {\frac {2}{5}}}$
$0={\sqrt {\frac {2}{5}}}\sin \left({\frac {2\pi x}{5}}\right)$
Use the form $\sin(kx)$ where $k=\left({\frac {2\pi }{5}}\right)$
$Period=\left({\frac {2\pi }{k}}\right)$
$Period=\left({\frac {2\pi }{\frac {2\pi }{5}}}\right)=5$
$\left({\frac {Period}{2}}\right)$ is the x-intercept
x-intercepts are 0, ${\frac {5}{2}}$, 5
0 and 5 are the edges of the box and ${\frac {5}{2}}$ is the only node.

#### Part 2

n$_{nodes}$=n-1
n$_{nodes}$=3-1=2
We're looking for two x-intercepts.
$0\div {\sqrt {\frac {3}{5}}}={\sqrt {\frac {3}{5}}}\sin \left({\frac {3\pi x}{5}}\right)\div {\sqrt {\frac {3}{5}}}$
$0={\sqrt {\frac {3}{5}}}\sin \left({\frac {3\pi x}{5}}\right)$
$k=\left({\frac {3\pi }{5}}\right)$ $Period=\left({\frac {3\pi }{\frac {3\pi }{5}}}\right)={\frac {10}{3}}$
x-intercepts are $\left({\frac {5n}{3}}\right)$
x-intercepts = 0, ${\frac {5}{3}},{\frac {10}{3}},5$
0 and 5 are the edges of the box, ${\frac {5}{3}}$ and ${\frac {10}{3}}$ are the nodes.
# Quantum Chemistry/Example 11
## Question

Use the 1D particle in a box model to estimate the wavelength of light required to excite an electron from a pi to pi* MO in ethene.

## Solution

The energy levels of a particle in a 1D box with a specific quantum number $n$, are as follows.
$E_{n}={\frac {h^{2}n^{2}}{8mL^{2}}}$
In this equation $h$ represents Planck's constant, $m$ is the mass of the particle, and $L$ is the length of the box.
The pi electron in the double bond between the carbon atoms in ethene can be approximated to the particle in a 1D box model. This means that the mass of the particle in this question will be the mass of an electron, and the length of the box corresponds to the bond length between the carbon atoms in the molecule ethene.
Additionally, the energy equation above needs to be transformed into a equation for $\Delta E$ since the electron is moving from one energy level to another.
$\Delta E=E_{f}-E_{i}={\frac {h^{2}n_{i}^{2}}{8mL^{2}}}-{\frac {h^{2}n_{f}^{2}}{8mL^{2}}}={\frac {h^{2}}{8mL^{2}}}(n_{f}^{2}-n_{i}^{2})$
The change in energy between the pi and pi* MO in ethene can now be calculated knowing that the bond length between doubly bonded carbon atoms is 133pm and the mass of an electron is 9.1093856x10$^{-31}$ kg. Moving from the ground state n=1 to an excited state of n=2 :
$\Delta E={\frac {h^{2}}{8mL^{2}}}(n_{f}^{2}-n_{i}^{2})$
$\Delta E={\frac {(6.626{\text{x}}10^{-34}{\frac {{\text{m}}^{2}{\text{kg}}}{\text{s}}})^{2}}{8(9.109\times 10^{-31}{\text{kg}})(133\times 10^{-12}{\text{m}})^{2}}}(2^{2}-1^{2})$
$\Delta E=1.0217\times 10^{-17}~{\text{J}}$
Now that the energy required to excite the electron to the pi* orbital is known, the wavelength of light can be calculated through the following equation, where c is the speed of light in a vacuum and $\lambda$ is the wavelength of light.
$\Delta E={\frac {hc}{\lambda }}$
The equation can then be re-arranged to solve for the wavelength of light.
$\lambda ={\frac {hc}{\Delta E}}$
By plugging in the known constants and the value for $\Delta E$ that has been calculated above, the wavelength can be found.
$\lambda ={\frac {hc}{\Delta E}}$
$\lambda ={\frac {(6.626\times 10^{-34}{\frac {{\text{m}}^{2}{\text{kg}}}{\text{s}}})(2.998\times 10^{8}{\frac {\text{m}}{\text{s}}})}{1.0217\times 10^{-17}{\text{J}}}}$
$\lambda =1.9\times 10^{-7}{\text{m}}$
$\lambda =19~{\text{nm}}$
Therefore the wavelength of light required to excite an electron from a pi to pi* molecular orbital in ethene is 19nm.
# Mass Media
## Table of contents

* Preface
* Introduction
* History
* Major Media Conglomerates

### Theories of media

* Influence of Media
* Ethics of Media
* Future of Media

### Mass media forms
#### Visual Media

* Television
* Posters
* Billboards
* Cinema

#### Audio media

* Radio

#### Print media

* Books
* Newspapers
* Magazines
* Essays
* Reviews

#### Internet media

* World Wide Web, Newsgroups and Discussion Groups
* Digital Television

### Working with media

* Journalism
* Editing
* Design and Advertising
* Independent Magazines (zines)
* Web Design

### Learning more about media

* Resources
* Organizations
* Courses

### Appendices

* Bibliography
* Glossary
* Authors and Contributors

## External links

   * Winston, Brian: Adventures in Cybersound: How are Media Born?
# Gmail/Send mail

Overview — Introduction — Facts — Basics — Advanced — Last tips — Other
Need to send an email?
Gmail makes it very easy to send emails to your friends. First log into your Gmail account. Once you have logged into your account, you will notice a prominent link under the Gmail logo that reads, "+ Compose". The "+ Compose" link is the link you will be pressing whenever you send an email.
Analyzing the screen
Once you click the "Compose Mail" link within your Gmail account, the page will change. You will be taken to a new page where you will actually write your email, or better said, compose your email. Let’s analyze this new page.
First you will notice that there are three new buttons that have appeared. These buttons are in the following order: Send, Save Now, and Discard. The Send button is only pressed once you have completed composing your email in order to send the email to the addressee. The Save Now button is used to save the email to your account's draft folder in order to access it, and maybe finish writing the email, in a later time. The Discard button is used when you would like to discard the email or better said, delete the email. Take note that when you discard an email it is deleted and unrecoverable.
Next we will notice there is a To: field on the page. This field is where you type the email address of the person you are sending the email to. For example, if you are sending an email to Linus and his email is linus@linux.org then you will type linus@linux.org into the To: field. You can also send the same email to multiple email address by adding a ; between email address. For example, I can send an email to more than one friend by typing their emails into the To: field in the following format: linus@linux.org; richardstallman@gnu.org
The next text field that we see is the Subject field. This is where you type the subject of the email. The subject will be seen by the person that you are sending the email prior to he or she opening the email. It is good practice to include descriptive but short subjects whenever possible.
Last but not least, we have the body field. This field is not labeled as body but that is what it is called. The body field is where you will be writing your email.
How to send the email
Once you have filled in each field you will be able to send the email. This is done by pressing the Send button which is located at the bottom of the screen. Try to send an email to your self in order to practice the fine art of email composing.
# Using Chrome/Introduction

Google Chrome is a cross-platform web browser developed by Google. It was first released in 2008 for Microsoft Windows, built with free software components from Apple WebKit and Mozilla Firefox. Versions were later released for Linux, macOS, iOS, and also for Android, where it is the default browser. The browser is also the main component of ChromeOS, where it serves as the platform for web applications. The current version is 109.0.5414.x, which was released on January 10, 2023.

## History of Chrome

The browser was first publicly released, officially as a beta version, on September 2, 2008, for Windows XP and newer, and with support for 43 languages, and later as a "stable" public release on December 11, 2008. The first official Chrome OS X and Linux developer previews were announced on June 4, 2009, with a blog post saying they were missing many features and were intended for early feedback rather than general use. In December 2009, Google released beta versions of Chrome for OS X and Linux. Google Chrome 5.0, announced on May 25, 2010, was the first stable release to support all three platforms.
# A Practical Guide To Mizar/Relations and Functions
## Relations

For this section, this will be the environment used:
A relation is just a set consisting only of ordered pairs:
where x,y,z are objects. For the rest of this section let P,Q,R denote relations. Then we can redefine P = R and P c= R as follows:
We can also infer that the intersection and set difference of a relation with a set is a relation again:
Similarly, the union of two relations is a relation again.
Exercise 1: Prove cluster P \/ R -> Relation-like;.
Note that the Cartesian product of two sets is of course a relation as well:
Of course there are functors to get the set of the first or second coordinates of a set containing pairs:
proj is short for projection. If the set X is already a relation, we call the projection to the first coordinate the domain and the projection to the second coordinate the range (or codomain or image) of the relation:
The notation construct is useful to avoid functor definitions like e.g.
notation has the advantage that we don't need to reference a definition for Mizar to recognize the equality.
RELAT_1 (as well as XTUPLE_0) contains several statements regarding domain and range for relations, for example:
Exercise 2: Find P,R such that the subset predicate in RELAT_1:13 becomes proper. I.e. prove ex P,R st rng(P /\ R) <> rng P /\ rng R. You can use RELAT_1:9.
The field of a relation is the union of its domain and range:
Exercise 3: Prove field(P \/ R) = field P \/ field R (RELAT_1:18) using XTUPLE_0:23, 27 and the commutativity (in-built) and associativity (XBOOLE_1:4) law of the set union.

## Exercise Solutions

Exercise 1:
Exercise 2:
Exercise 3: