OpenLLM-France/wikimedia huggingface dataset:
# Beginning Mathematics/Set Theory

← Logic and Deductive Reasoning | Number Theory →
In this section we shall go over some basic set theory.

# Sets

A set is a well defined collection of things; which things can be objects, numbers, symbols and so on. If we can determine whether the given thing belongs to the collection or not,then that collection is said to be well defined. Those things that make up a set are its members, and are also called elements of that set. Elements of sets may be objects, letters, items, or even other sets.
Sets can be finite(those containing a definite number of elements), or infinite(those containing an endless list of members).
Sets are denoted by capital letters of the English alphabet such as A,W,D,C,V,X,Z... . The elements of the set are indicated between the flower braces (also known as curl brackets),usually with members separated by commas. Ex: A={2,4,6,8,10}. Since all the elements are listed, it is called List form. This is also called Roster form.
Often, sets will have a pattern of elements or the elements will be similar in some fashion. At times, when it is clear what the pattern is, the set will end with "..." indicating the elements continue in that fashion. Ex:X={1,3,5,7,9,11,...}. Other times the set may be described in words or mathematical notation. Mathematical notation is read as in the following example: for the set X X={1,3,5,7,9,11,...}. This is shown as X={x/x is an odd number}
To denote an element is in a set we write: $x\in A$ this reads "x is an element of the set A" or "x is in A".

## Special Sets

Some sets are used frequently in mathematics so they are given by special notation.
$\mathbb {N} =\{Set\;of\;natural\;numbers\}=\{1,2,3,4,5,6,...\}$
$\mathbb {Z} =\{Set\;of\;all\;integers\}=\{...,-3,-2,-1,0,1,2,3,...\}$
$\mathbb {Q} =\{Set\;of\;rational\;numbers\}=\{{\frac {a}{b}}\;|\;a,b\in \mathbb {Z} \;and\;b\neq 0\}$
$\mathbb {R} =\{Set\;of\;real\;numbers\}$
$\mathbb {C} =\{Set\;of\;complex\;numbers\}=\{a+bi\;|\;a,b\in \mathbb {R} \;and\;where\;i^{2}=-1\}$

## Examples

$\{1,2,3\}$
$\{7,12,80\}$
$\{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}=\{{\text{letters of english alphabet}}\}$
$\{1,2,3,4,5,...\}=\{{\text{Set of positive integers}}\}$
$\{2,4,6,8,10,12,14...\}=\{{\text{Set of positive even integers}}\}=\{x\mid x\in \mathbb {Z} \;{\text{and }}x{\text{ is even}}\}$

## Basic operations
### Union

The union of two sets is denoted by $\bigcup$. The union of two sets is best described as "everything in both the sets once".
Formally: $A\bigcup B=\{x|x\in A\;or\;x\in B\}$
For example:
$A=\{1,2,3\}$ $B=\{4,5,6\}$
$A\bigcup B=\{1,2,3\}\bigcup \{4,5,6\}=\{1,2,3,4,5,6\}$

### Intersection

The intersection of two sets is denoted by $\bigcap$. The intersection of two sets is best described as "only what is in both of the sets".
Formally: $A\bigcap B=\{x|x\in A\;and\;x\in B\}$
For example:
$A=\{1,2,3,4,5,6\}$ $B=\{2,4,6,8,10\}$
$A\bigcap B=\{1,2,3,4,5,6\}\bigcap \{2,4,6,8,10\}=\{2,4,6\}$
# Belarusian/Appendix A
## Belarusian (Slavic)

| Translation | Phrase | Pronunciation (IPA) | Pronunciation (approximate) |
| Belarusian | Беларускі | /bʲelaˈruskʲi/ | (bye-lah-ROOH-skee) |
| hello | прывітанне | /prɪvʲiˈtanʲnʲe/ | (pree-vee-TAHN-nye) |
| good-bye | да пабачэння | /da pabaˈʧenʲnʲa/ | (duh puh-buh-CHEN-nyuh) |
| please | калі ласка | /kaˈʎi ˈlaska/ | (kuh-LEE LUS-kuh) |
| thank you | дзякуй | /ˈʥakuj/ | (DZYUH-kooy) |
| sorry | выбачайце | /vɪbaˈʧajʨe/ | (vee-buh-CHUY-tsye) |
| that one | той, тая, тое | /toj/, /ˈtaja/, /ˈtoje/ | (TOY, TUY-uh, TOY-eh) |
| how much? | колькі? | /ˈkoʎkʲi/ | (KOL-kee) |
| yes | так | /tak/ | (tuhk) |
| no | не | /ɲe/ | (nye) |
| I don’t understand | Не разумею | /ɲe razuˈmʲeju/ | (nye ruh-zoo-MEY-ooh) |
| where's the bathroom? | дзе туалет? | /ʥe tuaˈʎet/ | (dzye too-uh-LET) |
| generic toast | за здароў'е | /za zdaˈrou̯je/ | (zuh zduh-ROW-yeh) |
| Do you speak English? | Ці размаўляеце па-англійску? | /ʨi razmauˈʎajeʨe pa anɡˈʎijsku/ | (tsee ruz-muw-LYUY-et-eh puh un-GLEE-skoo) |
# Basic Algebra/Quadratic Functions/Chapter Review

Lesson 1. Graphing a Parabola
Lesson 2. Using the Discrimnant to Help Graph a Parabola
Lesson 3. Finding the Axis of Symmetry of a Parabola
Lesson 4. Finding the Vertex of a Parabola
Lesson 5. Graphing Quadratic Inequalities
# Basic Algebra/Radical Expressions and Equations/Chapter Test

Please post your test problems this way:
* Problem:(answer,level of difficulty(easy, medium, hard))
* Example 2x=6: (x=3,easy)
# Basic Algebra/Proportions and Proportional Reasoning/Solving Similar Triangles using Proportions
## Vocabulary

Proportions - Proportions are equations resulting from two or more ratios. (Ex. 2/3 = 4/x)
Similar triangles - have equal corresponding angles and proportional corresponding sides (Ex. If triangle ABC has side lengths 1, square root of 3, and 2, and angles of 30$^0$, 60$^0$, and 90$^0$, and triangle DEF has side lengths 2, 2*square root of 3, and 4, and angles of 30$^0$, 60$^0$, and 90$^0$, then triangles ABC and DEF are similar, because their corresponding angles are equal, and their corresponding sides are proportional to each other.)

## Example Problems

Proportions:
2/3 = 4/x Cross-multiply: 2x = 12 Divide both sides by 2: x = 6
Similar Triangles: If Triangles ABC and DEF are similar:
Find DE if AB=3, BC=4, EF=8. Since AB/DE = BC/EF, 3/DE = 4/8 Solving the proportion: DE=6 Find angle C when angle F=60$^0$. Since corresponding angles are congruent in similar triangles, angle C=angle F=60$^0$
# Basic Algebra/Radical Expressions and Equations/Chapter Review

Lesson 1. Square Root Functions
Lesson 2. Equations with Radicals
Lesson 3. Rational Exponents
Lesson 4. The Pythagorean Theorem
Lesson 5. The Distance Formula
# Basic Algebra/Proportions and Proportional Reasoning/Percents
## Vocabulary

Percent: Parts per one hundred

## Lesson

Suppose we had a grid of 100 squares, of which 17 are shaded in. There are several ways to express this as a ratio, as you learned in the previous lesson, such as ${\frac {17}{100}}$, the number of shaded squares compared to the number of squares in total. However, we can also write this as a percent. Since there are 17 shaded squares and 100 total, we say that $17\%$ of the squares are shaded. The $\%$ symbol stands for "percent".
Now suppose that instead of 100 squares, we have 50 squares, with 9 of them shaded. The percent shaded would not be $9\%$, because percent means "per 100", and we have 9 shaded squares out of 50 total squares. To find the percent, we need some number over 100, so we can set up a proportion.
          ${\frac {9}{50}}={\frac {x}{100}}$
Cross-multiplying gives us $9\times 100=50x$, and $x=18$.
We also could have noticed that $50\times 2=100$, so we just need to multiply the numerator, 9, by 2 to get our answer, 18.

## Practice Problems

Use / as the fraction line and put spaces between the wholes and fractions!
# Basic Algebra/Quadratic Functions/Chapter Test

Please post your test problems this way:
* Problem:(answer,level of difficulty(easy, medium, hard))
* Example 2x=6: (x=3,easy)
# Basic Algebra/Rational Expressions and Equations/Adding and Subtracting When the Denominators are Different
## Lesson

If you have to add two rational fractions with different denominators, as the first step, you have to find the LCM:
Now divide the LCM by both denominators and multiply by their respectives numerators:
The sum of the two results would be the new nominator:
This is another example:
We factorize both denominators and find the LCM
Now we divide and multiply:
We add the results to obtain the nominator; the denominator is the LCM:
We can factorize the nominator to simplify the result:

## Practice Problems

Easy
Medium
Hard
# Basic Algebra/Rational Expressions and Equations/Adding and Subtracting When the Denominators are the Same
## Lesson

You have to leave the same denominator and add the nominators:

## Practice Problems

Easy
Medium
Hard
# Basics of fine-art photography/Creating ruins photographs

Ruins photography is a genre (or sub-genre) of photographic art, the main theme of which is the depiction of abandoned, partially destroyed structures.

## General information

This genre, which overlaps with architectural and landscape photography, is relatively recent. Popular among some photographers in the West, where sometimes referred to as "ruin porn". In Russia, the study of abandoned objects is engaged in stalkers, but they sometimes don't have a camera and set themselves other goals.
The main subject of photography is abandoned buildings of recent past: hospitals, bomb shelters, barracks, hotels, industrial plants, automobile cemeteries, prisons, etc. A side theme is the "mastering" of abandoned buildings by nature (see photo right). Historical monuments of architecture can sometimes serve as the subject of the genre, but only as an exception.

## Ways of implementation

The process has similarities to Architectural photography, but if you are going to photograph underground objects or bunkers, you will need a flash, sometimes a powerful one, depending on the size of the object being photographed.
About where and how to find "photogenic" ruins, it is difficult to say: it depends on your location and willingness to overcome distances. Exept this you may have to buy additional equipment. As an option — to become a traveler, practicing hobo tourism, where sometimes you have to spend the night, or even live for a while in the ruins of abandoned houses.

## Interesting fact

Abandoned buildings sometimes serve as a backdrop for creating works in nude photography, what can be classified as the intersection of the two genres.

## Gallery

* Darul Aman Palace (Afghanistan) Photo taken on an Afghan expedition
* A dilapidated gatehouse on Eua Island (Tonga) Photo taken on the Expedition to Oceania
* An abandoned building in Kuala Lumpur (Malaysia) Photo taken on a Papuan expedition
* Abandoned structure near Tupisa (Bolivia) Photo taken on a Latin American expedition
* "Apartment" on the island of Tongatapu (Tonga) Photo taken on the Expedition to Oceania

## References

* ↑ Heide Rampetzreiter. "Ruinen-Fotografie: Detroit liegt auch in Europa" (in German). diepresse.com. Retrieved April 23, 2023.
# Basics of fine-art photography/Creating landscape photographs

Landscape is a genre of photography, where the main subject is virgin nature, — usually water bodies or earth surface.

## A little bit of theory

Sometimes details not related (or indirectly related) to nature may be present in the frame, supplementing the content. For example: an African hut against a savannah background; a boat (floating on an Asian river) with a fisherman in asian conical hat; a wild animal against a jungle background, etc. But, in any case, these elements should not overlap a significant part of the backdrop.

## Where to begin?

Creating works in this genre of photography is the easiest and most accessible way to gain first experience for beginners, as it does not require special skills required for studio shooting; another advantage is that no special accessories and additional equipment are required: tripod, flash, etc. Moreover, in our time of digital technology, a beginner will not face such difficulty as setting the exact shutter speed and aperture in difficult weather conditions, for example: snow-covered space combined with an overcast sky. In the recent past, when there was no alternative to analog photography, an inexperienced practitioner could ruin film in one session.

## About lighting

Front and side sunlight should be used. Don't take pictures in the midday light. Perhaps with the advent of experience you will learn to get around this problem, or even adapt it to your goals, but this will be later.
Personally, I like time before sunset , when the sun's rays seem to glide along the surface of the ground. But with such a shooting, the photo turns out to be yellowish, which not everyone appreciates positively.
To prevent side light from entering the lens (resulting in glare in the image), you need to use a lens hood.
Small errors can be corrected later with the help of graphic editors on a computer; big mistakes are uncorrectable.

## Gallery

* Lake Khövsgöl (Mongolia) Photo taken in Creative trip to Mongolia
* El Sillar (Bolivia) Photo taken on a Latin American expedition
* Cunene River (Namibia) Photo taken on a Third creative trip to Africa
* Bamyan (Afghanistan) Photo taken on an Afghan expedition
* Tufuvai Beach ('Eua, Tonga) Photo taken on the Expedition to Oceania
# Basics of fine-art photography/Creating works in a genre photography

Genre photography is the skill of depicting seemingly simple everyday scenes from an artistic perspective.

## General information

Genre photography and street photography can be considered related concepts, but the former came into being a little earlier. While there are many similarities between these genres, there are also differences: the former focuses on the subject and the latter — on the visual side of the photograph.

## Tips

In choosing this genre, you should be mobile, agile, observant and has compositional vision.
Each of your works should encourage the viewer to reflect on what came after, and perhaps before, what he see in the photo. If there is no storyline, the photo belongs to a different genre.

## More information

Ihere are also some similarities between genre photography and reportage photography, with the former necessarily being artistic while the latter may not be so: to illustrate newspaper articles, for example, the images may have minor flaws. This should be borne in mind when you choosing genre.

## Gallery

* The look (Afghanistan) Photo taken on an Afghan expedition
* An exasperated woman (Ethiopia) Photo taken on first African trip
* Dishwasher (Egypt) Photo taken on first African trip
* Stopover (Uganda) Photo taken on the second African trip
* Who settled in the pyramid? (Sudan) Photo taken on the third African trip
# Basic Writing/xml/bookinfo

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# Basics of fine-art photography/Creating works in a documentary photography

Documentary photography is a direction of photographic art based on the reproduction of real events.

## General information

The genre includes the depiction of moments of everyday life, historical events, natural disasters, etc. It partly overlaps with reportage photography, but there are some differences, one of them — the more complex storyline. It also has some points of contact with ethnographic, street and genre photography; used in photojournalism, amateur and art photography.
Documentary photography is not limited to creating works with people in the frame, and may combine natural objects, fauna and inanimate objects, but in any case, the works of supporters of this branch of photographic art are emotional and related to human emotions.

## Ways of implementation

It is impossible to give practical advice on how to achieve certain successes in documentary filming. In this case, the author's skill and practical skills do not take precedence over chance, and the time and place of events are sometimes difficult to predict. Always have a handy camera at hand, and luck will smile on you sooner or later. Better still, travel around the planet practicing hobo tourism, where adventures are always plentiful.
# Basics of fine-art photography/Creating stylized retro photo works

Retro style photography is an author's work that artificially imitates photographic images created many years ago.

## About the genre and methods of implementation

The genre has no clear temporal boundaries of the displayed theme. Photography can be imitating any period, — from invention of photocamera to, for example, the 1980s; it is much more difficult to recreate a stylized picture of the ancient period or primitive communal system, but theoretically possible
You can achieve the retro effect by artificially wear out the image, but initially you need to photograph a scene "from the past", in which should be present, for example, household items of bygone times, people dressed old-fashioned with once popular hairstyles...
To transform the original photo works, use graphic editors that have ready-made filters, or (if they are not available) use special textures of an old faded paper, scratched metal etc.

## Need to be aware

According to Wikipedia, retro is a style, but “retro photography” is already a genre.
Currently, there are no data on photographers who specialize exclusively in this genre

## Gallery

A series of photographs taken by author of the textbook in China (April, 2013) during a long expedition that included 9 countries.
* A man in hanfu
* Zhouzhuang — Chinese Venice
* Woman in hanfu
* Zhouzhuang — Chinese Venice
* Guard in ancient Chinese attire

## External links

* "The Free Dictionary (retro- )". thefreedictionary. Retrieved April 23, 2023.
# Four Knights Game: Grob Gambit

Interesting move. White sacrifices a pawn.
Four Knights Game: Grob Gambit :
| | a | b | c | d | e | f | g | h | |
| 8 | | | | | | | | | 8 |
| 7 | | | | | | | | | 7 |
| 6 | | | | | | | | | 6 |
| 5 | | | | | | | | | 5 |
| 4 | | | | | | | | | 4 |
| 3 | | | | | | | | | 3 |
| 2 | | | | | | | | | 2 |
| 1 | | | | | | | | | 1 |
| | a | b | c | d | e | f | g | h | |

## Black's responses

* 4...Nxg4 Black accepts the pawn sacrifice.After 5. Rg1 Nf6 6. d4 exd4 7. Nxd4 White has compensation for the sacrificed pawn.
* 4...d5 strong move. Black fights for the center.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Nc3/3...g6/4. d4/4...exd4/5. Nd5
## 4. d4 exd4 5. Nd5 - Three Knights, Steinitz, Steinitz-Rosenthal Variation

The move order 4. d4 exd4 5. Nd5 after 3...g6 is the Steinitz-Rosenthal Variation and is a line for White which sacrifices a pawn for aggressive play in the center and active development.

## 5...Bg7 line

The move 5...Bg7 by black ignores White's aggressive knight on d5 and decides to fianchetto their bishop on the kingside, just as planned by Black's third move, g6.

## 5...Nf6 line

With black's fifth move 5...Nf6 black aims to trade White's strong knight for black's normal knight. It also attacks the e4 pawn, which forces white to trade the knights now or a couple of moves later.
1. e4 e5 2. Nf3 Nc6 3. Nc3 g6 4. d4 exd4 5. Nd5
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Nxe5/3...Nxe5
## 3...Nxe5 - Irish Gambit Accepted

Black captures the free knight and gains a big material advantage. However, they might still get caught by surprise if they do not focus or play too passively.
1.e4 e5 2.Nf3 Nc6 3.Nxe5 Nxe5
# Winawer Defense
### 3...f5

The Winawer Defense, also known as the Gothic Defense, is an excellent response by Black instead of transposing into a Four Knight's Game. Black gambits the pawn on f5; if white accepts, the gambit creates an equalizing position for black and black often plays 4...Nf6.

## Theory table

For explanation of theory tables, see theory table and for notation, see algebraic notation.
1. e4 e5 2. Nf3 Nc6 3. Nc3 f5
| Winawer Defense | 4 Bb5 fxe4 | 5 Nxe4 Nf6 | +/= |
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Nc3/3...g6
## Theory table

For explanation of theory tables, see theory table and for notation, see algebraic notation.
1. e4 e5 2. Nf3 Nc6 3. Nc3 g6
| 4 | 5 | 6 |
| d4 | |
| Bc4 | |
When contributing to this Wikibook, please follow the Conventions for organization.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. b4
## 3. b4 - Pachman (Wing) Gambit

In response to Nc6, White plays b4 to develop their queen's bishop and create potential threats on the long diagonal. This move also emphasizes White's control of the center and indirectly challenges Black's knight on c6, potentially forcing it to move and creating weaknesses in Black's position. However, b4 also creates a potential weakness on the b-file that Black can exploit. By playing b4, White is taking a somewhat aggressive approach that can create opportunities for attack but also requires careful attention to the consequences of their moves. Also, Black can simply take the free pawn with Nxb4 or Bxb4. Pachman Wing Gambit starts with e4 e5 Nf3 Nc6 b4. The goal is to entice black to take the pawn and allow you to open the b-file. Aim to control the center and pressure the black king. Be mindful of defending the pawn on b4 if black declines the gambit, and avoid premature attacks. Play with precision and strategy in mind, as inexact moves can weaken your position. Pachman Wing Gambit can be tricky to counter, but certain strategies can help. Don't rush to capture the b4 pawn. Instead, aim to control the center and develop your pieces quickly. Try to anticipate your opponent's moves and have a solid defense in place. Keep an eye out for tactical opportunities and don't be afraid to counter-attack. Remember, playing accurate moves with patience and calm can often neutralize any gambit.
# Ponziani Opening
### 3. c3

The Ponziani opening is one of the oldest openings in chess history and chess books as it supports an immediate d4. Although it’s rarely seen in tournaments, it’s very underrated as white can control the centre and develop pieces quickly.
Some evaluations:
The defense 3. ...Nf6 often leads to a game where White has a small space advantage plus a small time advantage. Not really a bad deal at all. This defense, however, is rather quiet and drawish.
The defense 3. ... d5 leads to a good fight for both sides but really the person who knows current theory best will probably get a very good game.
The defense 3. ... f5 fairly often leads to a White advantage but again theory must be known.
The defense 3. ... d6 will usually lead to a positions often reached via the Ruy Lopez, Steinitz Variation, which are usually just fine for White.

## Theory table

For explanation of theory tables, see theory table and for notation, see algebraic notation. .
    1. e4 e5 2. Nf3 Nc6 3. c3
| | 4 | 5 | 6 | 7 |
| ... d5 | Qa4 f6 | Bb5 Nge7 | exd5 Qxd5 | O-O Bd7 | = |
| ... ... | Bb5 dxe4 | Nxe5 Qg5 | Qa4 Qxg2 | Bxc6 bxc6 | =+ |
| ... Nf6 | d4 d5 | Bb5 exd4 | e5 Ne4 | Nxd4 Bc5 | = |
| ... f5 | d4 fxe4 | Nxe5 Qf6 | Ng4 Qg6 | Bf4 d6 | += |
When contributing to this Wikibook, please follow the Conventions for organization.

## References

* Kasparov, Garry, & Keene, Raymond 1989 Batsford chess openings 2. ISBN 0-8050-3409-9.
* Nunn's Chess Openings. 1999. John Nunn (Editor), Graham Burgess, John Emms, Joe Gallagher. ISBN 1-8574-4221-0.
((Play the Ponziani)) Taylor/Hayward
* This Chess related article is a stub. Please assist Wikibooks by expanding it.
# Chess Opening Theory/1. d4/1...Nf6/2. c4/2...e5/3. Bd2
## 3. Bd2??

3. Bd2?? blunders a pawn.
# King's Indian, Orthodox
### 3. Nf3
## Theory table

For explanation of theory tables, see theory table and for notation, see algebraic notation. .
1. d4 Nf6 2. c4 d6 3. Nf3
| | 3 |
| King's Indian, Orthodox | ... g6 | Nc3 | +/= |
| | ... Nbd7 | Nc3 | +/= |
When contributing to this Wikibook, please follow the Conventions for organization.

## References

* Kasparov, Garry, & Keene, Raymond 1989 Batsford chess openings 2. ISBN 0-8050-3409-9.
* This Chess related article is a stub. Please assist Wikibooks by expanding it.
# Budapest Gambit

The move 2...e5 introduces the Budapest Gambit and is an energetic gambit. At first glance it seems to lose a pawn, however the pawn if taken is normally returned so that White avoids tactical issues. The resulting positions can lead to White having a slender advantage of having the bishop pair.

## Theory table

For explanation of theory tables, see theory table and for notation, see algebraic notation.
1. d4 Nf6 2. c4 e5
| | 3 | 4 | 5 | 6 | 7 |
| Fajarowicz gambit | dxe5 Ne4 | Nf3 Nc6 | a3 d6 | Qc2 Bf5 | Nc3 Nxf2 | += |
| Adler variation | ... Ng4 | Nf3 Bc5 | e3 Nc6 | Be2 O-O | O-O Re8 | += |
| Rubinstein variation | ... ... | Bf4 Nc6 | Nf3 Bb4+ | Nbd2 Qe7 | e3 Ngxe5 | += |
| | ... ... | ... ... | ... ... | Nc3 Bxc3+ | bxc3 Qe7 | += |
| | ... ... | ... g5 | Bg3 Bg7 | Nf3 Nc6 | Nc3 Ngxe5 | += |
| Alekhine variation | ... ... | e4 Nxe5 | f4 Nec6 | a3 a5 | Be3 Na6 | = |
| | d5 Bc5 | e3 d6 | = |