Tuple

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Tuple

In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. There is only one 0-tuple, referred to as the empty tuple. An n-tuple is defined inductively using the construction of an ordered pair.

Mathematicians usually write tuples by listing the elements within parentheses "( )" and separated by commas; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Sometimes other symbols are used to surround the elements, such as square brackets "[ ]" or angle brackets "⟨ ⟩". Braces "{ }" are used to specify arrays in some programming languages but not in mathematical expressions, as they are the standard notation for sets. The term tuple can often occur when discussing other mathematical objects, such as vectors.

In computer science, tuples come in many forms. Most typed functional programming languages implement tuples directly as product types, tightly associated with algebraic data types, pattern matching, and destructuring assignment. Many programming languages offer an alternative to tuples, known as record types, featuring unordered elements accessed by label. A few programming languages combine ordered tuple product types and unordered record types into a single construct, as in C structs and Haskell records. Relational databases may formally identify their rows (records) as tuples.

Tuples also occur in relational algebra; when programming the semantic web with the Resource Description Framework (RDF); in linguistics; and in philosophy.

Etymology

The term originated as an abstraction of the sequence: single, couple/double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., n‑tuple, ..., where the prefixes are taken from the Latin names of the numerals. The unique 0-tuple is called the null tuple or empty tuple. A 1‑tuple is called a single (or singleton), a 2‑tuple is called an ordered pair or couple, and a 3‑tuple is called a triple (or triplet). The number n can be any nonnegative integer. For example, a complex number can be represented as a 2‑tuple of reals, a quaternion can be represented as a 4‑tuple, an octonion can be represented as an 8‑tuple, and a sedenion can be represented as a 16‑tuple.

Although these uses treat ‑uple as the suffix, the original suffix was ‑ple as in "triple" (three-fold) or "decuple" (ten‑fold). This originates from medieval Latin plus (meaning "more") related to Greek ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded"), as in "duplex".

Names for tuples of specific lengths

Further information: Numeral prefix

 

Tuple length, ${\displaystyle n}$	Name	Alternative names
0	empty tuple	null tuple / empty sequence / unit / none left
1	monuple	single / singleton / monad
2	couple	double / ordered pair / two-ple / twin / dual / duad / dyad / twosome
3	triple	treble / triplet / triad / ordered triple / threesome
4	quadruple	quad / tetrad / quartet / quadruplet
5	quintuple	pentuple / quint / pentad
6	sextuple	hextuple / hexad
7	septuple	heptuple / heptad
8	octuple	octa / octet / octad / octuplet
9	nonuple	nonad / ennead
10	decuple	decad / decade (antiquated)
11	undecuple	hendecuple / hendecad
12	duodecuple	dozen / duodecad
13	tredecuple	baker's dozen
14	quattuordecuple	double septuple
15	quindecuple	triple quintuple
16	sexdecuple	quadruple quadruple
17	septendecuple
18	octodecuple
19	novemdecuple
20	vigintuple
21	unvigintuple
22	duovigintuple
23	trevigintuple
24	quattuorvigintuple
25	quinvigintuple
26	sexvigintuple
27	septenvigintuple
28	octovigintuple
29	novemvigintuple
30	trigintuple
31	untrigintuple
32	duotrigintuple
33	tritrigintuple
40	quadragintuple
41	unquadragintuple
50	quinquagintuple
60	sexagintuple
70	septuagintuple
80	octogintuple
90	nongentuple
100	centuple
1,000	milluple	chiliad

Note that for ${\displaystyle n\geq 3}$, the tuple name in the table above can also function as a verb meaning "to multiply [the direct object] by ${\displaystyle n}$"; for example, "to quintuple" means "to multiply by 5". If ${\displaystyle n=2}$, then the associated verb is "to double". There is also a verb "sesquiple", meaning "to multiply by 3/2". Theoretically, "monuple" could be used in this way too.

Properties

The general rule for the identity of two n-tuples is

${\displaystyle (a_{1},a_{2},\ldots ,a_{n})=(b_{1},b_{2},\ldots ,b_{n})}$ if and only if ${\displaystyle a_{1}=b_{1},{\text{ }}a_{2}=b_{2},{\text{ }}\ldots ,{\text{ }}a_{n}=b_{n}}$.

Thus a tuple has properties that distinguish it from a set:

1. A tuple may contain multiple instances of the same element, so
   tuple ${\displaystyle (1,2,2,3)\neq (1,2,3)}$; but set ${\displaystyle \{1,2,2,3\}=\{1,2,3\}}$.
2. Tuple elements are ordered: tuple ${\displaystyle (1,2,3)\neq (3,2,1)}$, but set ${\displaystyle \{1,2,3\}=\{3,2,1\}}$.
3. A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.

Definitions

There are several definitions of tuples that give them the properties described in the previous section.

Tuples as functions

The ${\displaystyle 0}$-tuple may be identified as the empty function. For ${\displaystyle n\geq 1,}$ the ${\displaystyle n}$-tuple ${\displaystyle \left(a_{1},\ldots ,a_{n}\right)}$ may be identified with the (surjective) function

${\displaystyle F~:~\left\{1,\ldots ,n\right\}~\to ~\left\{a_{1},\ldots ,a_{n}\right\}}$

with domain

${\displaystyle \operatorname {domain} F=\left\{1,\ldots ,n\right\}=\left\{i\in \mathbb {N} :1\leq i\leq n\right\}}$

and with codomain

${\displaystyle \operatorname {codomain} F=\left\{a_{1},\ldots ,a_{n}\right\},}$

that is defined at ${\displaystyle i\in \operatorname {domain} F=\left\{1,\ldots ,n\right\}}$ by

${\displaystyle F(i):=a_{i}.}$

That is, ${\displaystyle F}$ is the function defined by

${\displaystyle {\begin{alignedat}{3}1\;&\mapsto &&\;a_{1}\\\;&\;\;\vdots &&\;\\n\;&\mapsto &&\;a_{n}\\\end{alignedat}}}$

in which case the equality

${\displaystyle \left(a_{1},a_{2},\dots ,a_{n}\right)=\left(F(1),F(2),\dots ,F(n)\right)}$

necessarily holds.

Tuples as sets of ordered pairs

Functions are commonly identified with their graphs, which is a certain set of ordered pairs. Indeed, many authors use graphs as the definition of a function. Using this definition of "function", the above function ${\displaystyle F}$ can be defined as:

${\displaystyle F~:=~\left\{\left(1,a_{1}\right),\ldots ,\left(n,a_{n}\right)\right\}.}$

Tuples as nested ordered pairs

Another way of modeling tuples in Set Theory is as nested ordered pairs. This approach assumes that the notion of ordered pair has already been defined.

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set ${\displaystyle \emptyset }$.
2. An n-tuple, with n > 0, can be defined as an ordered pair of its first entry and an (n − 1)-tuple (which contains the remaining entries when n > 1):

   ${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},a_{3},\ldots ,a_{n}))}$

This definition can be applied recursively to the (n − 1)-tuple:

${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},(a_{3},(\ldots ,(a_{n},\emptyset )\ldots ))))}$

Thus, for example:

${\displaystyle {\begin{aligned}(1,2,3)&=(1,(2,(3,\emptyset )))\\(1,2,3,4)&=(1,(2,(3,(4,\emptyset ))))\\\end{aligned}}}$

A variant of this definition starts "peeling off" elements from the other end:

1. The 0-tuple is the empty set ${\displaystyle \emptyset }$.
2. For n > 0:

   ${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=((a_{1},a_{2},a_{3},\ldots ,a_{n-1}),a_{n})}$

This definition can be applied recursively:

${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=((\ldots (((\emptyset ,a_{1}),a_{2}),a_{3}),\ldots ),a_{n})}$

Thus, for example:

${\displaystyle {\begin{aligned}(1,2,3)&=(((\emptyset ,1),2),3)\\(1,2,3,4)&=((((\emptyset ,1),2),3),4)\\\end{aligned}}}$

Tuples as nested sets

Using Kuratowski's representation for an ordered pair, the second definition above can be reformulated in terms of pure set theory:

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set ${\displaystyle \emptyset }$;
2. Let ${\displaystyle x}$ be an n-tuple ${\displaystyle (a_{1},a_{2},\ldots ,a_{n})}$, and let ${\displaystyle x\rightarrow b\equiv (a_{1},a_{2},\ldots ,a_{n},b)}$. Then, ${\displaystyle x\rightarrow b\equiv \{\{x\},\{x,b\}\}}$. (The right arrow, ${\displaystyle \rightarrow }$, could be read as "adjoined with".)

In this formulation:

${\displaystyle {\begin{array}{lclcl}()&&&=&\emptyset \\&&&&\\(1)&=&()\rightarrow 1&=&\{\{()\},\{(),1\}\}\\&&&=&\{\{\emptyset \},\{\emptyset ,1\}\}\\&&&&\\(1,2)&=&(1)\rightarrow 2&=&\{\{(1)\},\{(1),2\}\}\\&&&=&\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\}\\&&&&\\(1,2,3)&=&(1,2)\rightarrow 3&=&\{\{(1,2)\},\{(1,2),3\}\}\\&&&=&\{\{\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\}\},\\&&&&\{\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\},3\}\}\\\end{array}}}$

n-tuples of m-sets

In discrete mathematics, especially combinatorics and finite probability theory, n-tuples arise in the context of various counting problems and are treated more informally as ordered lists of length n. n-tuples whose entries come from a set of m elements are also called arrangements with repetition, permutations of a multiset and, in some non-English literature, variations with repetition. The number of n-tuples of an m-set is mⁿ. This follows from the combinatorial rule of product. If S is a finite set of cardinality m, this number is the cardinality of the n-fold Cartesian power S × S × ⋯ × S. Tuples are elements of this product set.

Type theory

Main article: Product type

In type theory, commonly used in programming languages, a tuple has a product type; this fixes not only the length, but also the underlying types of each component. Formally:

${\displaystyle (x_{1},x_{2},\ldots ,x_{n}):{\mathsf {T}}_{1}\times {\mathsf {T}}_{2}\times \ldots \times {\mathsf {T}}_{n}}$

and the projections are term constructors:

${\displaystyle \pi _{1}(x):{\mathsf {T}}_{1},~\pi _{2}(x):{\mathsf {T}}_{2},~\ldots ,~\pi _{n}(x):{\mathsf {T}}_{n}}$

The tuple with labeled elements used in the relational model has a record type. Both of these types can be defined as simple extensions of the simply typed lambda calculus.

The notion of a tuple in type theory and that in set theory are related in the following way: If we consider the natural model of a type theory, and use the Scott brackets to indicate the semantic interpretation, then the model consists of some sets ${\displaystyle S_{1},S_{2},\ldots ,S_{n}}$ (note: the use of italics here that distinguishes sets from types) such that:

${\displaystyle [\![{\mathsf {T}}_{1}]\!]=S_{1},~[\![{\mathsf {T}}_{2}]\!]=S_{2},~\ldots ,~[\![{\mathsf {T}}_{n}]\!]=S_{n}}$

and the interpretation of the basic terms is:

${\displaystyle [\![x_{1}]\!]\in [\![{\mathsf {T}}_{1}]\!],~[\![x_{2}]\!]\in [\![{\mathsf {T}}_{2}]\!],~\ldots ,~[\![x_{n}]\!]\in [\![{\mathsf {T}}_{n}]\!]}$.

The n-tuple of type theory has the natural interpretation as an n-tuple of set theory:

${\displaystyle [\![(x_{1},x_{2},\ldots ,x_{n})]\!]=(\,[\![x_{1}]\!],[\![x_{2}]\!],\ldots ,[\![x_{n}]\!]\,)}$

The unit type has as semantic interpretation the 0-tuple.

Urban legend

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Urban_legend 

Colchester Overpass, also known as the "Bunnyman Bridge", outside Washington, D.C. near where the first hatchet attack ocurred

An urban legend (sometimes contemporary legend) is a genre of folklore comprising stories or fallacious claims circulated as true, especially as having happened to a "friend of a friend" or a family member, often with horrifying, humorous, or cautionary elements. These legends can be entertaining but often concern mysterious peril or troubling events, such as disappearances and strange objects or entities. Urban Legends may confirm moral standards, reflect prejudices, or be a way to make sense of societal anxieties. Urban legends in the past were most often circulated orally, but now can also be spread by any media. This includes newspapers, mobile news apps, e-mail, and most often, social media. Some urban legends have passed through the years with only minor changes. Generic Urban Legends are often altered to suit regional variations, but the lesson or moral remains the same.

Origin and structure

The term "urban legend", as used by folklorists, has appeared in print since at least 1968, when it was used by Richard Dorson. Jan Harold Brunvand, professor of English at the University of Utah, introduced the term to the general public in a series of popular books published beginning in 1981. Brunvand used his collection of legends, The Vanishing Hitchhiker: American Urban Legends & Their Meanings (1981) to make two points: first, that legends and folklore do not occur exclusively in so-called primitive or traditional societies, and second, that one could learn much about urban and modern culture by studying such tales.

Many urban legends are framed as complete stories with plot and characters. The compelling appeal of a typical urban legend is its elements of mystery, horror, fear, or humor. Often they serve as cautionary tales. Some urban legends are morality tales that depict someone acting in a disagreeable manner, only to wind up in trouble, hurt, or dead.

Urban legends will often try to invoke a feeling of disgust in the reader which tends to make these stories more memorable and potent. Elements of shock value can be found in almost every form of urban legend and are partially what makes these tales so impactful. An urban legend may include elements of the supernatural or paranormal.

Propagation and belief

As Jan Brunvand points out, antecedent legends including some of the motifs, themes and symbolism of the urtexts can readily be identified. Cases that may have been at least partially inspired by real events include "The Death Car" (traced by Richard Dorson to Michigan, United States); "the Solid Cement Cadillac" and the possible origin of "The Hook" in the 1946 series of Lovers' Lane murders in Texarkana, Texas, United States. The urban legend that Coca-Cola developed the drink Fanta to sell in Nazi Germany without public backlash originated as the actual tale of German Max Keith, who invented the drink and ran Coca-Cola's operations in Germany during World War II.

An example of a supposed ghost train, the The Silver Arrow ("Silverpilen")

The narrator of an urban legend may claim it happened to a friend (or to a friend of a friend), which serves to personalize, authenticate and enhance the power of the narrative whie distancing the teller from the tall tale. Many urban legends depict horrific crimes, contaminated foods, or other situations that would potentially affect many people. Anyone believing such stories might feel compelled to warn loved ones. On occasion, news organizations, school officials and even police departments have issued warnings concerning the latest threat. According to the "Lights Out" rumor, street-gang members would drive without headlights until a compassionate motorist responded with the traditional flashing of headlights, whereupon a prospective new gang-member would have to murder the citizen as a requirement of initiation. A fax retelling this legend received at the Nassau County, Florida, fire department was forwarded to police, and from there to all city departments. The Minister of Defence for Canada was taken in by it also; he forwarded an urgent security warning to all Ontario Members of Parliament.

Urban legends typically include common elements: the tale is retold on behalf of the original witness or participant; dire warnings are often given for those who might not heed the advice or lesson contained therein (a typical element of many e-mail phishing scams); and the tale is often touted as "something a friend told me", the friend being identified by first name only or not identified at all. Such legends seem to be believable and even provocative, as some readers are led in turn to pass them on, including on social media platforms that instantly reach millions worldwide. Many are essentially extended jokes, told as if they were true events.

Persistent urban legends do often maintain a degree of plausibility, as in the story a serial killer deliberately hiding in the back seat of a car. Another such example since the 1970s has been the recurring rumor that the Procter & Gamble Company was associated with Satan-worshippers because of details within its nineteenth-century "57" trademark. The legend interrupted the company's business to the point that it stopped using the trademark.

Relation to mythology

The earliest term by which these narratives were known, "urban belief tales", highlights what was then thought of as a key property: their tellers regarded the stories as true accounts, and the device of the FOAF (acronym for "Friend of a Friend" invented by English writer and folklorist Rodney Dale in 1976) was a spurious but significant effort at authentication. The coinage leads in turn to the terms "FOAFlore" and "FOAFtale". While at least one classic legend, the "Death Car", has been shown to have some basis in fact, folklorists have an interest in debunking those narratives only to the degree that establishing non-factuality warrants the assumption that there must be some other reason why the tales are told, re-told and believed. As in the case of myth, the narratives are believed because they construct and reinforce the worldview of the group within which they are told, or "because they provide us with coherent and convincing explanations of complex events".

Social scientists have started to draw on urban legends in order to help explain complex socio-psychological beliefs, such as attitudes to crime, childcare, fast food, SUVs and other "family" choices. The authors make an explicit connection between urban legends and popular folklore, such as Grimm's Fairy Tales, where similar themes and motifs arise. For that reason, it is characteristic of groups within which a given narrative circulates to vehemently reject claims or demonstrations of non-factuality; an example would be the expressions of outrage by police officers who are told that adulteration of Halloween treats by strangers (the subject of periodic moral panics) occurs extremely rarely, if at all.

Documentation

The Internet has made it easier both to spread and to debunk urban legends. For instance, the Usenet newsgroup alt.folklore.urban and several other websites, most notably snopes.com, focus on discussing, tracking, and analyzing urban legends. The United States Department of Energy had a now-discontinued service called Hoaxbusters that dealt with computer-distributed hoaxes and legends. The most notable such hoaxes are known as creepypastas, which are typically horror stories written anonymously. Although most are regarded as obviously false, some, such as the Slender Man, have gained a following of people that do believe in them.

Television shows such as Urban Legends, Beyond Belief: Fact or Fiction, and later Mostly True Stories: Urban Legends Revealed, feature re-enactments of urban legends, detailing the accounts of the tales and (typically later in an episode) revealing any factual basis they may have. The Discovery Channel TV show MythBusters (2003–2016) tried to prove or disprove several urban legends by attempting to reproduce them using the scientific method.

The 1998 film Urban Legend featured students discussing popular urban legends while at the same time falling victim to them.

Between 1992 and 1998 The Guardian newspaper "Weekend" section published the illustrated "Urban Myths" column by Phil Healey and Rick Glanvill, with content taken from a series of four books: Urban Myths, The Return of Urban Myths, Urban Myths Unplugged, and Now! That's What I Call Urban Myths. The 1994 comics anthology the Big Book of Urban Legends, written by Robert Boyd, Jan Harold Brunvand, and Robert Loren Fleming, featured 200 urban legends, displayed as comics.

The British writer Tony Barrell has explored urban legends in a long-running column in The Sunday Times. These include the story that Orson Welles began work on a Batman movie in the 1940s, which was to feature James Cagney as the Riddler and Marlene Dietrich as Catwoman; the persistent rumour that the rock singer Courtney Love is the granddaughter of Marlon Brando; and the idea that a famous 1970s poster of Farrah Fawcett contains a subliminal sexual message concealed in the actress's hair.

Genres

Crime

As with traditional urban legends, many internet rumors are about crimes or crime waves - either fictional or based on real events that have been largely exaggerated. Such stories can be problematic, both because they purport to be relevant modern news and because they do not follow the typical patterns of urban legends.

Medicine

Some legends are medical folklore, such as the claim that eating watermelon seeds will result in a watermelon growing in the stomach, or that going outdoors just after showering will result in catching a cold.

Internet

Internet urban legends are those spread through the internet, as through Usenet or email or more recently through other social media. They can also be linked to viral online content. Some take the form of chain letters and spread by e-mail, directing the reader to share them or to meet a terrible fate, and following a recognizable outline of hook, threat, and finally request. Others are fake virus-alerts, warning people of nonexistent threats to their computers, often appearing as online pop-ups claiming to be giveaways or store coupons.

Paranormal

Paranormal urban-legend stories usually involve someone encountering something supernatural, such as a cryptid—for instance, Bigfoot or Mothman, legendary creatures for which evidence is wanting but which have legions of believers. Research shows that people experiencing sudden or surprising events (such as a Bigfoot sighting) may significantly overestimate the duration of the event.

Marketing

Companies have been accused of hiding "secret messages" behind their logos or packaging, as in the case of the old Procter & Gamble symbol, supposedly an occult figure that gave panache to the brand. (If the thirteen stars in the symbol were connected a certain way, it would show three sixes in a row.) Similarly, a video of a Christian woman "exposing" Monster Energy for using the Hebrew alphabet symbol for the letter "M" to disguise the number 666 went viral on Facebook.

Some urban legends have been used intentionally for comic purpose in advertising. The most well-known examples include the use of a Sasquatch in Jack Link commercials, known as "Messin' with Sasquatch," and the use of unicorns in Icebreakers ads. Another is the New Jersey Devils hockey team, named for the state's popular cryptid, the Jersey Devil.