Mathematical fallacy

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof. For example, the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and should not be applied in the cases that are the exceptions to the rules.

Real Madrid C.F.

Real Madrid Club de Fútbol (Spanish pronunciation: [reˈal maˈðɾið ˈkluβ ðe ˈfuðβol]; Royal Madrid Football Club), commonly known as Real Madrid, or simply as Real, is a professional football club based in Madrid, Spain.

Founded in 1902 as Madrid Football Club, the team has traditionally worn a white home kit since inception. The word Real is Spanish for Royal and was bestowed to the club by King Alfonso XIII in 1920 together with the royal crown in the emblem. The team has played its home matches in the 81,044-capacity Santiago Bernabéu Stadium in downtown Madrid since 1947. Unlike most European sporting entities, Real Madrid's members (socios) have owned and operated the club throughout its history.

The club is the most valuable sports team in the world, worth €2.5 billion ($3.4 billion) and the world's highest-earning football club for 2013–14, with an annual revenue of €549.5 million. The club is one of the most widely supported teams in the world. Real Madrid is one of three founding members of the Primera División that have never been relegated from the top division, along with Athletic Bilbao and Barcelona. The club holds many long-standing rivalries, most notably El Clásico with Barcelona and the El Derbi madrileño with Atlético Madrid.

Real Betis

Real Betis Balompié, SAD, more commonly referred to as Real Betis (pronounced: [reˈal ˈβetis]) or just Betis, is a Spanish football club based in Seville, in the autonomous community of Andalusia. Founded on 12 September 1907, it plays in La Liga, having won the Segunda División in the 2014–15 season. It holds home games at Estadio Benito Villamarín in the south of the city.

Among other titles, the club – which in 1932 became the first Andalusian team to play in La Liga – won the league title in 1935 and the Copa del Rey in 1977 and 2005.

Maintaining an historic city rivalry with Sevilla FC, its motto is Viva el Betis manque pierda! ("Long live Betis even when they lose!").

History

The name is derived from Baetis, the Roman name for the Guadalquivir river which passes through Seville. Betis initially attracted support from the working classes although a large number of aristocrats, including the King of Spain also supported the team. Real was added in 1914 after the club received patronage from King Alfonso XIII.

Club Tijuana

Club Tijuana Xoloitzcuintles de Caliente, commonly referred to as Xolos de Tijuana, or simply as Xolos, is a Mexican professional football club from Tijuana, Baja California, Mexico. Founded in January 2007, the club was promoted to Liga MX in 2011, where they have played since. They won their first title in the 2012 Apertura.

History

The club is the latest in a long line of league teams in the city of Tijuana. Gallos Caliente was instituted in the summer of 2006. The team's name was later changed to Club Tijuana Xoloitzcuintles de Caliente. Team owner Jorge Hank Rhon announced the construction of the Estadio Caliente, a new stadium with a capacity for 33,333 people near Grupo Caliente's Agua Caliente Racetrack. Jorge Alberto Hank, the 28-year-old son of Jorge Hank Rhon, is the President of the team. They became the Apertura 2012 champions after defeating Toluca 4-2 in a two-legged series.

The team advanced to the Primera División de México with a win at home over Irapuato, 2–1 on May 21, 2011.

Chess notation

Chess notation is the term for several systems that have developed to record either the moves made in a game of chess or the position of pieces on a chessboard. The earliest systems of notation used lengthy narratives to describe each move; these gradually evolved into terser notation systems. Currently algebraic chess notation is the accepted standard and is widely used. Algebraic notation has several variations. Descriptive chess notation was used in English- and Spanish-language literature until the late 20th century, but is now obsolete. There are some special systems for international correspondence chess. Portable Game Notation is used when working with computer chess programs. Systems also exist for transmission using Morse code over telegraph or radio.

Recording moves

Some move-recording notations are designed mainly for use by human players; others are designed for use by computers.

Notation systems for humans

In recognized competitions, all players are required to record all the moves of both players in order to resolve disputes about whether a player has made an illegal move and what the position should now be. In addition, if there is a time limit rule that requires each player to complete a specified number of moves in a specified time, as there is in most serious competition, an accurate count of the moves must be kept. All chess coaches strongly recommend the recording of one's games so that one can look for improvements in one's play. The algebraic and descriptive notations are also used in books about chess.