Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree. The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite.

Definition

A set is a well defined collection of distinct objects. The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:

Potato chip

A potato chip (American English) or crisp (British English) is a thin slice of potato that has been deep fried, baked, kettle cooked, or popped until crunchy. Potato chips are commonly served as a snack, side dish, or appetizer. The basic chips are cooked and salted; additional varieties are manufactured using various flavorings and ingredients including herbs, spices, cheeses, and artificial additives.

"Crisps", however, may also refer to many different types of savory snack products sold in the United Kingdom and Ireland, some made from potato, but some made from corn, tapioca or other cereals, just as there are other varieties of chips in the United States.

Potato chips are a predominant part of the snack food market in Western countries. The global potato chip market generated total revenues of US$16.49 billion in 2005. This accounted for 35.5% of the total savory snacks market in that year ($46.1 billion).

History

In the UK, the origin of the potato chip has been attributed to English food writer William Kitchiner's 1822 cookbook The Cook's Oracle, which was a bestseller in England and the United States, and includes a recipe for "Potatoes fried in Slices or Shavings", which instructs readers to "peel large potatoes, slice them about a quarter of an inch thick, or cut them in shavings round and round, as you would peel a lemon; dry them well in a clean cloth, and fry them in lard or dripping". The earliest reference of the potato chip in the United States is in Mary Randolph's The Virginia House-Wife (1824), which includes a recipe explicitly derived from Kitchiner's earlier cookbook. Boston Housekeeper N.K.M. Lee's cookbook, The Cook's Own Book (1832), also contains a recipe for the potato chip that references Kitchiner's cookbook.

Set (music)

A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.

A set by itself does not necessarily possess any additional structure, such as an ordering. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called segments); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis.

Two-element sets are called dyads, three-element sets trichords (occasionally "triads", though this is easily confused with the traditional meaning of the word triad). Sets of higher cardinalities are called tetrachords (or tetrads), pentachords (or pentads), hexachords (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g.,), octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.

Set (darts)

A set in darts consists of a sequence of legs (games) played, ending when the count of legs won meets certain criteria. Throwing first is considered an advantage in a leg, so players alternate who throws first in each leg during the set.

Generally, a set will consist of the best of five legs (first player to win three) - although there are some exceptions. The most notable being the Winmau World Masters, where a set is the best of three legs (first to two).

Example

During the final of the 2007 PDC World Darts Championship, Raymond van Barneveld defeated Phil Taylor by seven sets to six in one of the most dramatic darts matches of all-time. The breakdown of how each set went is shown here.

Taylor won the first two sets by three legs to nil, then added the third set before van Barneveld took the next two. The match continued and went to six sets all. Neither player managed to gain a two-leg advantage in the final set "tiebreak" so the deciding leg was played when the score reached 5-5. Before the final leg, the players threw for the bullseye with the nearest given the advantage of throwing first.

Set construction

Set construction is the process by which a construction manager undertakes to build full scale scenery suitable for viewing by camera, as specified by a production designer or art director working in collaboration with the director of a production to create a set for a theatrical, film or television production. The set designer produces a scale model, scale drawings, paint elevations (a scale painting supplied to the scenic painter of each element that requires painting), and research about props, textures, and so on. Scale drawings typically include a groundplan, elevation, and section of the complete set, as well as more detailed drawings of individual scenic elements which, in theatrical productions, may be static, flown, or built onto scenery wagons. Models and paint elevations are frequently hand-produced, though in recent years, many Production Designers and most commercial theatres have begun producing scale drawings with the aid of computer drafting programs such as AutoCAD or Vectorworks.