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Set (mathematics)

In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written; the concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, can be used as a foundation from which nearly all of mathematics can be derived; the German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a well-defined collection of distinct objects; the objects that make up a set can be anything: numbers, letters of the alphabet, other sets, so on. Georg Cantor, one of the founders of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre: A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have the same elements. For technical reasons, Cantor's definition turned out to be inadequate; the most basic properties are that a set can have elements, that two sets are equal if and only if every element of each set is an element of the other. There are two common ways of describing, or specifying the members of, a set: roster notation and set builder notation.. These are examples of intensional definitions of sets, respectively; the Roster notation method of defining a set consist of listing each member of the set. More in roster notation, the set is denoted by enclosing the list of members in curly brackets: A = B =. For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified in roster notation as,where the ellipsis indicates that the list continues in according to the demonstrated pattern. In roster notation, listing a member does not change the set, for example, the set is identical to the set.

Moreover, the order in which the elements of a set are listed is irrelevant, so is yet again the same set. In set-builder notation, the set is specified as a subset of a larger set, where the subset is determined by a statement or condition involving the elements. For example, a set F can be specified as follows: F =. In this notation, the vertical bar means "such that", the description can be interpreted as "F is the set of all numbers n, such that n is an integer in the range from 0 to 19 inclusive". Sometimes the colon is used instead of the vertical bar. Set-builder notation is an example of intensional definition. Another method is by using a rule or semantic description: A is the set whose members are the first four positive integers. B is the set of colors of the French flag; this is another example of intensional definition. If B is a set and x is one of the objects of B, this is denoted as x ∈ B, is read as "x is an element of B", as "x belongs to B", or "x is in B". If y is not a member of B this is written as y ∉ B, read as "y is not an element of B", or "y is not in B".

For example, with respect to the sets A =, B =, F =, 4 ∈ A and 12 ∈ F. If every element of set A is in B A is said to be a subset of B, written A ⊆ B. Equivalently, one can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A; the relationship between sets established by ⊆ is called containment. Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B. If A is a subset of B, but not equal to B A is called a proper subset of B, written A ⊊ B, or A ⊂ B, or B ⊋ A; the expressions A ⊂ B and B ⊃ A are used differently by different authors. Examples: The set of all humans is a proper subset of the set of all mammals. ⊆. ⊆. There is a unique set with no members, called the empty set, denoted by the symbol ∅; the empty set is a subset of every set, every set is a subset of itself: ∅ ⊆ A. A ⊆ A. A partition of a set S is a set of nonempty subsets of S such that every element x in S is in one of these subsets; that is, the subsets are pairwise disjoint, the union of all the subsets of the partition is S.

The power set of a set S is the set of all subsets of S. The power set contains S itself and the empty set because these are both subsets of S. For example, the pow

Deodatus of Nevers

Deodatus of Nevers was a bishop of Nevers from 655. Deodatus lived with Arbogast in the monastery of Ebersheim, established by Childeric II near Sélestat in the forest of Haguenau. Deodatus' establishment of the monastery Juncturae in the present town of Saint-Dié followed his appointment as the bishop of Nevers, he placed Jointures under the Rule of Saint Columban. He baptized the son of Saint Hunna, named Deodatus and, venerated as a saint. Hunna's son became a monk at Ebersheim. After 664 Deodatus renounced his see to withdraw to the so-called valley of "Galilaea" in the Vosges, where he lived as a hermit in a cell. Tradition states that he died in the arms of bishop of Treves; the town of Saint-Dié grew up around the monastery of Jointures. However, some sources connect the name with Deodatus of Blois. "Deodatus of Nevers". Biographisch-Bibliographisches Kirchenlexikon. Deodatus von St. Dié Den hellige Deodatus av Nevers

Ola Salo

Ola Salo is the Swedish rock vocalist of the Swedish glam rock band The Ark. He lived in Rottne, Växjö Municipality, in Sweden when he was a child, where, in 1991, he and his friends Jepson and Leari started the band The Ark. Salo is bisexual. Salo and the other members of the band had an international breakthrough in 2000 with the album We Are The Ark, containing the signature song "It Takes a Fool to Remain Sane", a song Salo wrote after watching the Danish film Idioterne. In October 2006 during a party celebrating the new Swedish embassy in Washington, D. C; the Ark was performing on stage. As a plane was flying low overhead Salo said "In this country, you don't know where those planes are headed. Well, this one seems to be heading in the right direction anyway..." meaning the airport, but suddenly adding "...to the White House" which happened to be in the same direction. This caused controversy as many newspapers reported that Salo had "wished an airplane to crash into the White House". Salo said that it was a bad joke, "totally unserious way of being cheeky toward the White House" and not a political statement.

The band ended up cancelling its entire U. S. tour. On 10 March 2007, Salo and the band The Ark won Melodifestivalen 2007 and went on to represent Sweden in the Eurovision Song Contest 2007 with the song "The Worrying Kind", where, in which they came 18th with 51 points. In 2008, Salo translated Andrew Lloyd Webber's Jesus Christ Superstar into Swedish for a performance in Malmö where he played the role of Jesus, as a result he stated that "2008 will be a quiet year" for The Ark after a hectic 2007 with the Eurovision Song Contest and the release of Prayer for the Weekend. In 2009. Salo featured Come On You Preachers, he will participate in Stjärnornas stjärna which will be broadcast on TV4. In 2019, Salo opened his headline residency It takes a fool to remain sane in the Rondo, directed by Edward af Sillén