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Finite set

In mathematics, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle finish counting. For example, is a finite set with five elements; the number of elements of a finite set is called the cardinality of the set. A set, not finite is called infinite. For example, the set of all positive integers is infinite:. Finite sets are important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Formally, a set S is called finite for some natural number n; the number n is the set's cardinality, denoted as |S|. The empty set or Ø is considered finite, with cardinality zero. If a set is finite, its elements may be written — in many ways — in a sequence: x 1, x 2, …, x n. In combinatorics, a finite set with n elements is sometimes called an n-set and a subset with k elements is called a k-subset.

For example, the set is a 3-set – a finite set with three elements – and is a 2-subset of it. Any proper subset of a finite set S has fewer elements than S itself; as a consequence, there cannot exist a bijection between a finite set S and a proper subset of S. Any set with this property is called Dedekind-finite. Using the standard ZFC axioms for set theory, every Dedekind-finite set is finite, but this implication cannot be proved in ZF alone; the axiom of countable choice, a weak version of the axiom of choice, is sufficient to prove this equivalence. Any injective function between two finite sets of the same cardinality is a surjective function. Any surjection between two finite sets of the same cardinality is an injection; the union of two finite sets is finite, with | S ∪ T | ≤ | S | + | T |. In fact, by the inclusion–exclusion principle: | S ∪ T | = | S | + | T | − | S ∩ T |. More the union of any finite number of finite sets is finite; the Cartesian product of finite sets is finite, with: | S × T | = | S | × | T |.

The Cartesian product of finitely many finite sets is finite. A finite set with n elements has 2n distinct subsets; that is, the power set of a finite set is finite, with cardinality 2n. Any subset of a finite set is finite; the set of values of a function when applied to elements of a finite set is finite. All finite sets are countable; the free semilattice over a finite set is the set of its non-empty subsets, with the join operation being given by set union. In Zermelo–Fraenkel set theory without the axiom of choice, the following conditions are all equivalent: S is a finite set; that is, S can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number. S has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. S can be given a total ordering, well-ordered both forwards and backwards; that is, every non-empty subset of S has both a least and a greatest element in the subset.

Every one-to-one function from P into itself is onto. That is, the powerset of the powerset of S is Dedekind-finite; every surjective function from P onto itself is one-to-one. Every non-empty family of subsets of S has a minimal element with respect to inclusion. S can be well-ordered and any two well-orderings on it are order isomorphic. In other words, the well-orderings on S have one order type. If the axiom of choice is assumed the following conditions are all

María Cristina Muñoz y Borbón

María Cristina Muñoz y Borbón was a Spanish aristocrat, 1st Marchioness of la Isabela. Born on 19 April 1840 in the Royal Palace of Madrid, she was the uterine sister of Queen Isabella II, one the children conceived in the marriage between Queen Regent Maria Christina and her morganatic husband Agustín Fernando Muñoz, the 1st Duke of Riánsares, she was granted the titles of Viscountess of La Dehesilla and the title of Marchioness of La Isabela in 1847. She married in the Malmaison on 20 October 1860 José María Bernaldo de Quirós y González de Cienfuegos, Marquis of Campo Sagrado, whom with she had 3 surviving children: Jesús María, Ana Germana and María de los Desamparados, she died in Tutor Street 22, Madrid on 20 December 1921. She is an ancestor of Íñigo Méndez de Vigo and Carlos Espinosa de los Monteros


Carrouges is a commune in the Orne department in north-western France. The inhabitants are known as Carrougiennes; the town is home to a castle, built in the 14th century by Jean de Carrouges and restored after the Hundred Years War. In the Middle Ages, Carrouges was part of the deanery of Asnebec. In 1490, under King Charles VIII, Jean de Blosset, Lord of Carrouges and grand marshal of Normandy, founded on his land, with its castle, a college six canons payments the appointment of the lord, the collateur profits; these payments were estimated in 1698 to 200 pounds in Memory of the General Alençon by the intendant, M. de Pommereuil. It is stated in 1698, that a court called "breadbasket tax" was located in Carrouges. During the French Revolution of 1789, the town was called "Carrouges-la-Montagne". In August 1944, occupied by German forces, the city was liberated by the troops of the 3rd U. S. armored division, following the deeds of Mayor Geslain, who after being released from the German soldiers who had arrested him, indicated the positions of the Germans to the Americans.

Communes of the Orne department Parc naturel régional Normandie-Maine INSEE commune file

Tunde Nightingale

Earnest Olatunde Thomas, known as Tunde Nightingale or The Western Nightingale, was a Nigerian singer and guitarist, best known for his unique jùjú music style, following in the tradition of Tunde King. Born in Ibadan, he attended school in Lagos, served in the army, worked for a railway company, he formed his first group, a three piece band comprising guitar and shekere, in 1944. This was at the onset of the period Nigerian musicians began to use guitar as part of their recordings, but his juju style of music was not the most popular among the Lagos elite who dominated the social scene and performances were limited to bars which provided limited income opportunities. In 1952, his band under the name of Tunde Nightingale and His Agba Jolly Orchestra held regular performances at the West African Club, Ibadan, his contemporaries included I. K. Dairo and Dele Ojo. By 1952 his group had expanded to eight members, played at the West African Club in Ibadan. Between 1954 and 1964, Nightingale's popularity fared better.

In the mid-1960s, he signed a record deal with Mr Jossy Fajimolu, it was during this period his music gained popularity among some socialites in Lagos who deemed the music more suitable for social parties than dance halls. He developed his style of juju music that included room for praise and incantations as party music and played at halls or dances, his style of music was known as So Wàmbè a double entendre reference to the beads draped over the hips of dancing women. By the 1960s, his popularity had grown among Lagos socialites; when he returned, he signed with the TYC label. In all, he recorded over 40 albums in his career. Modern stars like Queen Ayo Balogun continue to be influenced by his style. Apart from the fact that he "sounded" like a nightingale, he kept a live bird in his home

Borophagus littoralis

Borophagus littoralis is an extinct species of the genus Borophagus of the subfamily Borophaginae, a group of canids endemic to North America from the early Miocene epoch through the Pliocene epoch. Borophagus littoralis existed for 18.4 million years. Borophagus littoralis was named by Cope in 1892 and is considered synonymous with Osteoborus diabloensis. Borophagus littoralis, like other Borophaginae, are loosely known as "bone-crushing" or "hyena-like" dogs. Though not the most massive borophagine by size or weight, it had a more evolved capacity to crunch bone than earlier, larger genera such as Epicyon, which seems to be an evolutionary trend of the group. During the Pliocene epoch, Borophagus began being displaced by Canis genera such as Canis edwardii and by Canis dirus. Early species of Borophagus were placed in the genus Osteoborus until but the genera are now considered synonyms. Borophagus littoralis led a hyena-like lifestyle scavenging carcasses of dead animals. Typical features of this genus are powerful jaws.

Its crushing premolar teeth and strong jaw muscles would have been used to crack open bone, much like the hyena of the Old World. The adult animal is estimated to have been about 80 cm in length, similar to a coyote, although it was much more powerfully built. Alan Turner, "National Geographic: Prehistoric Mammals", pp. 112–114. ISBN 0-7922-7134-3 Xiaoming Wang, "The Origin and Evolution of the Dog Family" Accessed 1/30/06. Picture of an Osteoborus skull in a museum, from "World of the Wolf." Russell Hunt, "Ecological Polarities Of the North American Family Canidae: A New Approach to Understanding Forty Million Years of Canid Evolution". Wang et al. "Phylogenetic Systematics of the Borophaginae." Bulletin of the American Museum of Natural History, No. 243, Nov. 17 1999

Avenir: Mirai

Avenir: Mirai is the third studio album of Hello! Project is her first album in Japanese. Concurrently, it was her last album, released when she was still a member of the idol group Morning Musume, it was released on December 29, 2004 with catalog number EPCE-2030. The album's first pressing included Hello! Project photo card #94. "Miraizu" Lyrics: Junko Sadou Composer: Rie Arranger: Chijou Maeno "Aruite Yukou... Mirai he" Lyrics: Kaori Iida Composer: Kazunori Ashizawa Arranger: Chijou Maeno "Door no Mukō de Bell ga Natteta" Lyrics: Tokuko Miura Composer: Tsunku Arranger: Kouji Makaino "Jounetsu no Tobira" Lyrics: Natsumi Watanabe Composer: Kazunori Ashizawa Arranger: Chijou Maeno "Shinjugai" Lyrics: Natsumi Watanabe Composer: Ichiro Kaneda Arranger: Chijou Maeno "Watashi no Naka ni Ite" Lyrics: Composer: Hitoshi Haba "Senritsu" Lyrics: Yuho Iwasato Composer: Taisei Arranger: Chijou Maeno "Aegekai ni Dakarete" Lyrics: Tokuko Miura Composer: Tsunku Arranger: Chijou Maeno "Sekai de Ichiban Kirei na Hoshizora" Lyrics: Yuho Iwasato Composer: Kiyonori Matsuo Arranger: Chijou Maeno "Mafuyu no Rondo" Lyrics: Junko Sadou Composer: Kazunori Ashizawa Arranger: Chijou Maeno "Sayonara Made ni Shitai 10 no Koto" Lyrics: Yuho Iwasato Composer: Hitoshi Haba Arranger: Chijou Maeno "Arifureta Kiseki" Lyrics: Junko Sadou Composer: Akira Inaba Arranger: Chijou Maeno Avenir: Mirai entry at Up-Front Works Official Website Avenir: Mirai entry at Kaori Iida Database