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.

Athletic Bilbao

Athletic Club; (Basque: Athletic Kluba), also commonly known as Athletic de Bilbao; (Basque: Bilboko Athletic Kluba), is a professional football club, based in Bilbao, Basque Country, Spain.

They are known as Los Leones (The Lions) because their stadium was built near a church called San Mamés (Saint Mammes). Mammes was a semi-legendary early Christian thrown to the lions by the Romans. Mammes pacified the lions and was later made a saint.

The club is one of three founding members of the Primera División that have never been relegated from the top division, the others being Real Madrid and Barcelona. Athletic have won the league on eight occasions and in the historical classification of La Liga, Athletic are in fourth place. In the table of Copa del Rey titles, Athletic is second only to Barcelona, having won the Cup 23 times. The club also has a women's team, which has won four championships in the Primera División Femenina.

The club is known for its cantera policy of bringing young Basque players through the ranks, as well as recruiting top Basque players from other clubs (like Joseba Etxeberria and Javi Martínez). Athletic official policy is signing professional players native to or trained in football in the greater Basque Country, including Biscay, Gipuzkoa, Álava and Navarre (in Spain); and Labourd, Soule and Lower Navarre (in France). Since its foundation, Athletic has played exclusively with players meeting the criteria to be deemed as Basque players, and has been one of the most successful teams of La Liga. This can be seen as a unique case in European football; it has gained Athletic both admirers and critics. The club has been praised for promoting home grown players and club loyalty. Athletic is one of only three professional clubs in Spain in La Liga (the others being Real Madrid, and Barcelona) that is not a sports corporation; it is owned and operated by its club members.

Copa del Rey

The Copa del Rey (English: King's Cup) is an annual football cup competition for Spanish football teams. Its full name is Campeonato de España – Copa de Su Majestad el Rey (Championship of Spain – Cup of His Majesty the King), and is commonly known as La Copa.

The competition was founded in 1903, thus making it the oldest Spanish football competition. Typically, the winner of Copa del Rey goes to the UEFA Europa League; unless they qualify for the UEFA Champions League.

Barcelona are the current holders, who won their 27th Copa del Rey against Athletic Bilbao at the 2015 Copa del Rey Final held at the Camp Nou.

History

In 1902, a competition under the name Copa de la Coronación, was played after Carlos Padrós, later president of Real Madrid, suggested a football tournament to celebrate the coronation of King Alfonso XIII. Four other teams joined Madrid FC for the competition: FC Barcelona, Club Español de Foot-Ball, New Foot-Ball de Madrid and Club Bizcaya (a team made up of players from Athletic Club and Bilbao FC) which eventually defeated Barcelona in the final. That cup is on display in the Athletic Bilbao museum and the club includes the victory in its honours list. Nevertheless, it is considered only the forerunner of the Copa del Rey and the Royal Spanish Football Federation officially doesn't recognize it.

Bilbao

Bilbao (/bɪlˈbaʊˌ -ˈbɑːoʊ/;Spanish: [bilˈβao]; Basque: Bilbo [bilβo]) is a municipality and city in Spain, a major city in the province of Biscay in the autonomous community of the Basque Country. It is the largest municipality of the Basque Country and the tenth largest in Spain, with a population of 353,187 in 2010. The Bilbao metropolitan area has roughly 1 million inhabitants, making it one of the most populous metropolitan areas in northern Spain; with a population of 875,552 the comarca of Greater Bilbao is the fifth-largest urban area in Spain. Bilbao is also the main urban area in what is defined as the Greater Basque region.

Bilbao is situated in the north-central part of Spain, some 16 kilometres (10 mi) south of the Bay of Biscay, where the estuary of Bilbao is formed. Its main urban core is surrounded by two small mountain ranges with an average elevation of 400 metres (1,300 ft).

After its foundation in the early 14th century by Diego López V de Haro, head of the powerful Haro family, Bilbao was a commercial hub of the Basque Country that enjoyed significant importance in Green Spain. This was due to its port activity based on the export of iron extracted from the Biscayan quarries. Throughout the nineteenth century and the beginning of the twentieth, Bilbao experienced heavy industrialisation, making it the centre of the second-most industrialised region of Spain, behind Barcelona. At the same time an extraordinary population explosion prompted the annexation of several adjacent municipalities. Nowadays, Bilbao is a vigorous service city that is experiencing an ongoing social, economic, and aesthetic revitalisation process, started by the iconic Bilbao Guggenheim Museum, and continued by infrastructure investments, such as the airport terminal, the rapid transit system, the tram line, the Alhóndiga, and the currently under development Abandoibarra and Zorrozaurre renewal projects.

Barcelona

Barcelona (/bɑːrsəˈloʊnə/, Catalan: [bəɾsəˈlonə], Spanish: [barθeˈlona]) is the capital city of the autonomous community of Catalonia in Spain and Spain's second most populated city, with a population of 1.6 million within its administrative limits. Its urban area extends beyond the administrative city limits with a population of around 4.7 million people, being the seventh-most populous urban area in the European Union after Paris, London, Madrid, the Ruhr area, Berlin and Milan. It is the largest metropolis on the Mediterranean Sea, located on the coast between the mouths of the rivers Llobregat and Besòs, and bounded to the west by the Serra de Collserola mountain range, the tallest peak of which is 512 metres (1,680 ft) high.

Founded as a Roman city, in the Middle Ages Barcelona became the capital of the County of Barcelona. After merging with the Kingdom of Aragon, Barcelona continued to be an important city in the Crown of Aragon as an economical and administrative center of this Crown and the capital of the Principality of Catalonia. Besieged several times during its history, Barcelona has a rich cultural heritage and is today an important cultural center and a major tourist destination. Particularly renowned are the architectural works of Antoni Gaudí and Lluís Domènech i Montaner, which have been designated UNESCO World Heritage Sites. The headquarters of the Union for the Mediterranean is located in Barcelona. The city is known for hosting the 1992 Summer Olympics as well as world-class conferences and expositions and also many international sport tournaments.