In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions and computer science Computer science or computing science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe, and transform information. Computer science, a tuple is an ordered list of elements. In set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, an (ordered) n-tuple is a sequence In mathematics, a sequence is an ordered list of objects . Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence. A sequence is a (or ordered list) of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively A recursive definition or inductive definition is one that defines something in terms of itself , albeit in a useful way. For it to work, the definition in any given case must be well-founded, avoiding an infinite regress using the construction of an ordered pair In mathematics, an ordered pair is a collection of objects having two coordinates , such that one can always uniquely determine the object, which is the first coordinate (or first entry or left projection) of the pair as well as the second coordinate (or second entry or right projection). If the first coordinate is a and the second is b, the usual. Tuples are usually written by listing the elements within parentheses '( )' and separated by commas; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Sometimes other delimiters are used, such as brackets '[ ]' or angle brackets '⟨ ⟩'. Braces '{ }' are almost never used for tuples, as they are the standard notation for sets A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented 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.

Tuples are often used to describe other mathematical objects. In algebra Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure, for example, a ring In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element. To qualify as a ring, the set together with its two operations must satisfy certain conditions—namely, the set must be an abelian group under addition and a monoid under is commonly defined as a 3-tuple (E,+,×), where E is some set, and '+', and '×' are functions The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain from the Cartesian product In mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept E×E to E with specific properties. In computer science, tuples are directly implemented as product types in most functional programming languages. More commonly, they are implemented as record types In computer science, a record is one of the simplest data structures, consisting of two or more values or variables stored in consecutive memory positions; so that each component (called a field or member of the record) can be accessed by applying different offsets to the starting address, where the components are labeled instead of being identified by position alone. This approach is also used in relational algebra Relational algebra, an offshoot of first-order logic , deals with a set of finitary relations (see also relation (database)) which is closed under certain operators. These operators operate on one or more relations to yield a relation. Relational algebra is a part of computer science.

Origin of name

The term originated as an abstraction of the sequence: single, double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., n‑tuple, ... The unique 0‑tuple is called the null tuple. A 1‑tuple is called a singleton, a 2‑tuple is called a pair and a 3‑tuple is a triple or triplet. The n can be any nonnegative integer. For example, a complex number A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the can be represented as a 2‑tuple, a quaternion In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A striking feature of quaternions is that the product of two quaternions is noncommutative, meaning that the product of two can be represented as a 4‑tuple, an octonion In mathematics, the octonions are a nonassociative and noncommutative extension of the quaternions. Their 8-dimensional normed division algebra over the real numbers is the widest possible that can be obtained from the Cayley–Dickson construction. The octonion algebra is often denoted O, or in blackboard bold by can be represented as an octuple, (many mathematicians write the abbreviation "8‑tuple") and a sedenion In abstract algebra, sedenions form a 16-dimensional non-associative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions. The set of sedenions is denoted by can be represented as a 16‑tuple.

Although these uses treat ‑tuple as the suffix, the original suffix was ‑ple as in "decuple", ten‑fold. This originates from a medieval Latin suffix ‑plus, "more", related to Greek ‑πλοῦς, which replaced the classical and late antique ‑plex "folded".^[1]

Formal definitions

Characteristic properties

The main properties that distinguish a tuple from, for example, a set A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented 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 are that

1. it can contain an object more than once;
2. the objects appear in a certain order;
3. it has finite size.

Note that (1) distinguishes it from an ordered set and that (2) distinguishes it from a multiset In mathematics, a multiset is a generalization of a set. While each member of a set has only one membership, a member of a multiset can have more than one membership (meaning that there may be multiple instances of a member in a multiset, not that a single member instance may appear simultaneously in several multisets). The term "multiset&. This is often formalized by giving the following rule for the identity of two n-tuples:

(a₁, a₂, …,a_n) = (b₁, b₂, …, b_n) if and only if a₁ = b₁, a₂ = b₂, …, and a_n = b_n.

Tuples as functions

An n-tuple can also be regarded as a function The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain whose domain is the natural numbers In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century { 1, 2, …, n } (or { 0, 1, …, n-1 }); that is, a set of index-element pairs:

(a₁, a₂, …,a_n) ≡ { (1, a₁), (2, a₂), … (n, a_n) }

or

(a₀, a₁, …,a_n−1) ≡ { (0, a₀), (1, a₁), … (n−1, a_n−1) }.

Tuples as nested ordered pairs

Another way of formalizing tuples is as nested ordered pairs In mathematics, an ordered pair is a collection of objects having two coordinates , such that one can always uniquely determine the object, which is the first coordinate (or first entry or left projection) of the pair as well as the second coordinate (or second entry or right projection). If the first coordinate is a and the second is b, the usual. Namely,

1. the 0-tuple (i.e. the empty tuple) is represented by the empty set Ø;
2. an n-tuple, with n > 0, can be defined as an ordered pair In mathematics, an ordered pair is a collection of objects having two coordinates , such that one can always uniquely determine the object, which is the first coordinate (or first entry or left projection) of the pair as well as the second coordinate (or second entry or right projection). If the first coordinate is a and the second is b, the usual of its first entry and an (n−1)-tuple containing the remaining entries:

   (a₁, a₂, …, a_n) = ( a₁, (a₂, …, a_n-1, a_n)).

Thus, for example, the tuple (3, 5, 3) would be the same as (3,(5,(3,Ø))).

This definition mirrors the most common representation of tuples as linked lists In computer science, a linked list is a data structure that consists of a sequence of data records such that in each record there is a field that contains a reference to the next record in the sequence — as used, for example, in standard implementations of the Lisp programming language Lisp is a family of computer programming languages with a long history and a distinctive, fully parenthesized syntax. Originally specified in 1958, Lisp is the second-oldest high-level programming language in widespread use today; only Fortran is older. Like Fortran, Lisp has changed a great deal since its early days, and a number of dialects have.

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

1. the 0-tuple is the empty set Ø;
2. for n > 0,

(a₁, a₂, …, a_n) = ((a₁, a₂, …, a_n-1), a_n).

Thus, for example, the tuple (3, 5, 3) would be the same as (((Ø,3),5),3).

Tuples as nested sets

Using Kuratowski's representation for an ordered pair In mathematics, an ordered pair is a collection of objects having two coordinates , such that one can always uniquely determine the object, which is the first coordinate (or first entry or left projection) of the pair as well as the second coordinate (or second entry or right projection). If the first coordinate is a and the second is b, the usual, the second definition above can be reformulated in terms of pure set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics as:

1. the 0-tuple (i.e. the empty tuple) is represented by the empty set Ø;
2. if x is an n-tuple, and a is any element, then { {x}, {x,a} } is an (n + 1)-tuple.

In this formulation, the tuple (3, 5, 3) would be

{ { (3, 5) }, { (3, 5), 3 } } =

{ { { { (3) }, { (3), 5 } } }, { { { (3) }, { (3), 5 } }, 3 } } =

{ { { { { { Ø }, { Ø, 3 } } }, { { { Ø }, { Ø, 3 } }, 5 } } }, { { { { { Ø }, { Ø, 3 } } }, { { { Ø }, { Ø, 3 } }, 5 } }, 3 } }

Relational model

In database theory Database theory encapsulates a broad range of topics related to the study and research of the theoretical realm of databases and database management systems, the relational model The relational model for database management is a database model based on first-order predicate logic, first formulated and proposed in 1969 by E.F. Codd uses a definition similar with tuples as functions above, but each tuple element is identified by a distinct name, called an attribute, instead of a number. A tuple in the relational model is formally defined as a finite function The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain that maps attributes to values. Its purpose is the same as in mathematics, that is, to indicate that an object consists of certain components, but the components are identified by name instead of position, which leads to a more user-friendly In design, Usability is the study of the ease with which people can employ a particular tool or other human-made object in order to achieve a particular goal. This can include endevours as varied as consumer electronics, communication, and knowledge transfer objects and mechanical objects such as a door handles or a hammer and practical notation,^[2] for example:

( player : "Harry", score : 25 )

In this notation, attribute–value pairs may appear in any order. The distinction between tuples in the relational model and those in set theory is only superficial. Imposing an arbitrary total order In set theory, a total order, linear order, simple order, or ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain on the attributes, e.g. player ≤ score in the above example, and then ignoring the attribute names yields a 2-tuple. Conversely, a 2-tuple may be interpreted as relational model tuple over the attributes {1, 2}.^[2]

In the relational model, a relation In SQL, a database language for relational databases, a relation variable is called a table is a (possibly empty) finite set of tuples all having the same finite set of attributes, which is more formally called the sort of the relation, or just referred to as column names.^[2] A tuple is usually implemented as a row In the context of a relational database, a row—also called a record or tuple—represents a single, implicitly structured data item in a table. In simple terms, a database table can be thought of as consisting of rows and columns or fields. Each row in a table represents a set of related data, and every row in the table has the same structure in a database table, but see relational algebra Relational algebra, an offshoot of first-order logic , deals with a set of finitary relations (see also relation (database)) which is closed under certain operators. These operators operate on one or more relations to yield a relation. Relational algebra is a part of computer science for means of deriving tuples not physically represented in a table.

Type theory

In type theory In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general. In programming language theory, a branch of computer science, type theory can refer to the design, analysis and study of type systems, although some computer, commonly used in programming languages A programming language is an artificial language designed to express computations that can be performed by a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine, to express algorithms precisely, or as a mode of human communication, a tuple has a product type; this fixes not only the length, but also the underlying types of each component. Formally (x₁, ...,x_n) : T₁×...×T_n, and the projections are term constructors π₁(x) : T₁, ..., π_n(x) : T_n. The tuple with labeled elements used in the relational model (see section above) has a record type. Both of these types can be defined as simple extensions of simply typed lambda calculus The simply typed lambda calculus is a typed interpretation of the lambda calculus with only one type constructor: that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical uses of the untyped.^[3]

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 In mathematics, model theory is the study of mathematical structures such as groups, fields, graphs, or even universes of set theory, using tools from mathematical logic. A structure that gives meaning to the sentences of a formal language is called a model for the language. If a model for a language moreover satisfies a particular sentence or of a type theory, and use the Scott brackets to indicate the semantic interpretation, then the model consists of some sets T₁, ..., T_n (note: the use of italics here that distinguishes sets from types) such that T₁ = T₁, ..., T_n = T_n, and the interpretation of the basic terms is x₁ T₁, ..., x_n T_n. The type theory tuple has the natural interpretation as a set theory n-tuple: (x₁, ...,x_n) = (x₁, ...,x_n).^[4] The unit type In the area of mathematical logic, and computer science known as type theory, a unit type is a type that allows only one value . The carrier (underlying set) associated with a unit type can be any singleton set. There is an isomorphism between any two such sets, so it is customary to talk about the unit type and ignore the details of its value has as semantic interpretation the 0-tuple.

See also

• Arity In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the number of domains in the corresponding Cartesian product. The term springs from such words as unary, binary, ternary, etc
• Exponential object In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. An exponential object may also be called a power object or map object
• Formal language A formal language is a set of words, i.e. finite strings of letters, symbols, or tokens. The set from which these letters are taken is called the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar ; accordingly, words that belong to a formal language are sometimes called well-formed words (
• OLAP: Multidimensional Expressions
• Relation (mathematics) In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations
• Tuplespace

References

1. ^ OED, s.v. "triple", "quadruple", "quintuple", "decuple"
2. ^ ^{a b c} Serge Abiteboul, Richard Hull, Victor Vianu, Foundations of databases, Addison-Wesley, 1995, ISBN 0201537710, p. 29–33
3. ^ Pierce, Benjamin (2002). Types and Programming Languages. MIT Press. pp. 126–132. ISBN 0-262-16209-1.
4. ^ Steve Awodey, From sets, to types, to categories, to sets, 2009, preprint A preprint is a draft of a scientific paper that has not yet been published in a peer-reviewed scientific journal

The set theory definitions herein are found in any textbook on the topic, e.g.

• Gaisi Takeuti, W. M. Zaring, Introduction to Axiomatic Set Theory, Springer GTM Graduate Texts in Mathematics is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow books of a standard size. This particular series is easily identified by a white band at the top of the book 1, 1971, ISBN 978-0-387-90024-7, p. 14
• Abraham Adolf Fraenkel, Yehoshua Bar-Hillel, Azriel Lévy, Foundations of set theory, Elsevier Studies in Logic Vol. 67, Edition 2, revised, 1973, ISBN 0720422701, p. 33
• Keith Devlin Keith J. Devlin is an English mathematician and popular science writer, The Joy of Sets. Springer Verlag, 2nd ed., 1993, ISBN 0-387-94094-4, pp. 7-8
• George J. Tourlakis, Lecture Notes in Logic and Set Theory. Volume 2: Set theory, Cambridge University Press, 2003, ISBN 978-0-521-75374-6, pp. 182-193

External links

• Prefixes & Words Based On Latin Number Names

Categories: Data management Categories: Computer data | Data | Project management | Information retrieval | Information technology management | Mathematical notation | Sequences and series In mathematics, a sequence is a list of objects which have been ordered in a sequential fashion; such that each member either comes before, or after, every other member. More formally, a sequence is a function with a domain equal to the set of positive integers | Basic concepts in set theory | Type theory

See Also:

What Links Here:

Recently Viewed Pages:

• Home
• Sitemap
• Spiritual Retreats: Blogs
• Resorts: Blogs
• Blog Search
• Dictionary Search
• Carpooling: Images
• Venezuela: Blogs
• Directories: Blogs
• Peru: Answers

Most Popular Pages:

• Resorts: Blogs
• English Language: Blogs
• Spiritual Retreats: Blogs
• Venezuela: Blogs
• Peru: Answers
• Baedeker: Images
• Directories: Blogs
• Book: Answers
• Scar Literature: Answers
• Directory Sites

Related 'Tuple' Searches: