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.

UEFA Europa League

The UEFA Europa League, previously called the UEFA Cup, is an annual association football club competition organized by UEFA since 1971 for eligible European football clubs. Clubs qualify for the competition based on their performance in their national leagues and cup competitions.

Previously called the UEFA Cup, the competition has been known as the UEFA Europa League since the 2009–10 season, following a change in format. For UEFA footballing records purposes, the UEFA Cup and UEFA Europa League are considered the same competition, with the change of name being simply a rebranding.

In 1999, the UEFA Cup Winners' Cup was abolished and merged with the UEFA Cup. For the 2004–05 competition a group stage was added prior to the knockout phase. The 2009 re-branding included a merge with the UEFA Intertoto Cup, producing an enlarged competition format, with an expanded group stage and changed qualifying criteria. The winner of the UEFA Europa League qualifies for the UEFA Super Cup, and from the 2014–15 season onwards the winner of the UEFA Europa League also qualifies for the next edition of the UEFA Champions League. The winner enters at least at the play-off round (assuming they do not already qualify through domestic performance), and will enter the group stage if the berth reserved for the Champions League title holders is not used.

A.C. Milan

Associazione Calcio Milan (Italian pronunciation: [assotʃatˈtsjoːne ˈkaltʃo ˈmiːlan]), commonly referred to as A.C. Milan or simply Milan, is a professional Italian football club based in Milan, Lombardy, that plays in Serie A. Milan was founded in 1899 by English lace-maker Herbert Kilpin and businessman Alfred Edwards among others. The club has spent its entire history, with the exception of the 1980–81 and 1982–83 seasons, in the top flight of Italian football, known as Serie A since 1929–30.

With 18 officially recognised UEFA and FIFA titles, they are the second most successful club in the world in terms of number of international titles, together with Boca Juniors and Real Madrid, and behind Al Ahly with twenty titles. Milan has won a joint record of three Intercontinental Cups and one of its successor, the FIFA Club World Cup. Milan have also won the European Cup/Champions League on seven occasions, second only to Real Madrid. They have also won the UEFA Super Cup a joint record five times and the Cup Winners' Cup twice. Milan has won every major competition in which it has competed, with the exception of the Europa League (in this competition they have lost two semi-finals in 1972 and in 2002). Domestically, with 18 league titles Milan is the joint-second most successful club in Serie A behind Juventus (31 titles), along with local rivals Inter. They have also won the Coppa Italia five times, as well as six Supercoppa Italiana triumphs.

Ricardo Rodriguez

Ricardo Rodriguez or Ricardo Rodríguez may refer to:

Milan

Milan (English /mᵻˈlæn/ or US /məˈlɑːn/;Lombard, Milanese variant: Milan [miˈlã]),Italian: Milano [miˈlaːno]), the second-most populous city in Italy, is the capital of Lombardy. The city proper has a population of about 1.3 million, while its urban area (the 5th-largest in the EU) comprises an estimated 5 million people (former Provinces of Milan and Monza-Brianza, with other Comuni included in the former Province of Varese). The enormous suburban sprawl that followed the post-war boom of the 1950s–1960s has resulted in a polycentric metropolitan area, known as Greater Milan, of 7 to 10 million people, stretching over the provinces of Milan, Bergamo, Como, Lecco, Lodi, Monza and Brianza, Pavia, Varese and Novara. The Milan metropolitan region is part of the so-called Blue Banana, the area of Europe with the highest population and industrial density. In terms of GDP, Milan has the third largest economy among EU cities (after London and Paris) and the largest among European non-capital cities.