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Encyclopedia > Tuple

In mathematics, a tuple is a finite sequence (also known as an "ordered list") of objects, each of a specified type. A tuple containing n objects is known as an "n-tuple". For example the 4-tuple (or "quadruple"), with components of respective types PERSON, DAY, MONTH and YEAR, could be used to record that a certain person was born on a certain day of a certain month of a certain year. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...


Tuples are used to describe mathematical objects that consist of specified components. For example, a directed graph is defined as a tuple (V, E) where V is the set of nodes and E is a subset of V × V that denotes the edges. The type of the first object is "set of nodes" and the type of the second is "set of edges". This article just presents the basic definitions. ...

Contents

Names of tuples

The term originated as an abstraction of the sequence: single, double, triple, quadruple, quintuple, n-tuple. A tuple of length n is usually described as an n-tuple. A 2-tuple is called a pair; a 3-tuple is a triple or triplet. The n can be any nonnegative integer. For example, a complex number can be represented as a 2-tuple, and a quaternion can be represented as a 4-tuple. Further constructed names are possible, such as octuple, but many mathematicians find it quicker to write "8-tuple", even if still pronouncing this "octuple". In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...


Names for tuples of specific length

  • Empty tuple (0)
  • Single (1) (or singleton or sole or only - many others)
  • Double (2) (or pair or twice)
  • Triple (3) (or triplet or treble or thrice or threesome or troika or trio)
  • Quadruple (4)
  • Quintuple or Pentuple (5)
  • Sextuple or Hextuple (6)
  • Septuple (7)
  • Octuple (8)
  • Nonuple (9)
  • Decuple (10)
  • Centuple (100)

Formal definitions

The main properties that distinguish a tuple from, for example, a set are that In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...

  1. it can contain an object more than once, and
  2. the objects appear in a certain order.

Note that (1) distinguishes it from an ordered set and that (2) distinguishes it from a multiset. This is often formalized by giving the following rule for the identity of two n-tuples: ... In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ...

(a1, a2, …,an) = (b1, b2, …, bn) ↔ a1 = b1, a2 = b2, …, an = bn.

Since a n-tuple is indexed by the numbers 1…n (or 0…n-1), it can be regarded as a function from a subset of : Look up Function in Wiktionary, the free dictionary. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

(a1, a2, …,an) ≡ fa: ℕn → A: iai.

Another way of formalizing tuples is by mapping them to more primitive constructs in set theory such as ordered pairs. For example, an n-tuple (with n > 2) can be defined as an ordered pair of its first entry and an (n−1)-tuple containing the remaining entries: Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ... In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ...

(a1, a2, …, an) = (a1, (a2, …, an))

Using the usual set-theoretic definition of an ordered pair and letting the empty set represent the empty tuple, this results in the following inductive definition: In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ...

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

Using this definition, (1,2,2) would be

(1,(2,(2,))) = (1,(2, {{2}, {2, }} )) = (1, {{2}, {2, {{2}, {2, }}}} ) = {{1}, {1, {{2}, {2, {{2}, {2, }}}}}}

There is an important similarity here with the way Lisp originally used the ordered pair abstraction to inductively create all of its n-tuple and list structures: Lisp is a family of computer programming languages with a long history and a distinctive fully-parenthesized syntax. ...

  1. a special symbol NIL represents the empty list;
  2. if X is a list and A an arbitrary value then the pair (A X) represents a list with the head (i.e. first element) A and the tail (i.e. the remainder of the list without the head) X.

Usage in computer science

In computer science, tuple can have three distinct meanings. Typically in functional and some other programming languages, a tuple is a data object that holds several objects, similar to a mathematical tuple. Such an object is also known as a record. Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... In computer science, object composition (not to be confused with function composition) is a way and practice to combine simple objects or data types into more complex ones. ...


The Eiffel programming language has a built-in notion of tuple. The type Eiffel is an object-oriented programming language which emphasizes the production of robust software. ...

 TUPLE [X, Y, Z] 

has, as its values, tuples of three or more elements, of which the first is of type X, the second of type Y and the third of type Z. This can also be written with tags:

 TUPLE [tag1: X, tag2: Y, tag3: Z] 

without affecting the resulting type. An actual tuple, corresponding to this type, is written in bracket notation, for example

 [x1, y1, z1] 

with x1 of type X etc. If t is such a tuple, its elements can be accessed, in the form using tags, as t.tag1 etc.; they can also be set in the same way, as in t.tag2 := y2 which replaces the second element, of type Y, by y2. A value of type TUPLE [X, Y, Z] can be assigned to a variable of the same type but also to one of type TUPLE [X, Y], or TUPLE [X], or just TUPLE which covers all tuples. This is thanks to the definition that TUPLE [X, Y], for example, covers sequences of at least (rather than exactly) two elements, with the first two of the types given. Tuple types fit well in an object-oriented context, where they save writing a class when all you need is simple sequences of values with associated access and set mechanisms for each field.


In some languages, and especially in database theory, a tuple is defined as a finite function that maps field names to a certain value. Its purpose is the same as in mathematics, namely to indicate that a certain entity or object consists of certain components and/or has certain properties, but here these components are identified by a unique field name and not by a position, which often leads to a more user-friendly notation. The general term for this construct is an associative array; other programming languages have yet other names for the concept. A database is an information set with a regular structure. ... An associative array (also map, hash, dictionary, finite map, lookup table, and in query-processing an index or index file) is an abstract data type composed of a collection of keys and a collection of values, where each key is associated with one value. ...


A small example of a tuple would be:

( player : "Harry", score : 25 )

which is a function that maps the field name "player" to the string "Harry" and the field name "score" to the number 25. Note that the order of the components is not relevant, so the same tuple can also be written as:

( score : 25, player : "Harry" )

In the relational model such tuples are typically used to represent a single proposition; in this case there exists a player with the name "Harry" and a score of 25.


In programming languages, tuples are used to form data structures. For example, the following could be a structure that represents a node in a doubly linked list: In computer science, a linked list is one of the fundamental data structures used in computer programming. ...

( value : 16, previous-node : 1174782, next-node : 1174791 )

See also

In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes... In mathematics, logic, and computer science, a formal language is a set of finite-length words (i. ... Eiffel is an object-oriented programming language which emphasizes the production of robust software. ... In computer science, Linda is a parallel programming language which is implemented as an extension of other (sequential) languages such as Prolog, C, or Java. ... Multidimensional Expressions (MDX) is a query language for OLAP databases, much like SQL is a query language for relational databases. ... In mathematics, a finitary relation is defined by one of the formal definitions given below. ... This article may be too technical for most readers to understand. ... A unit type is a mathematical type that allows only one value. ...

External links

  • Prefixes & Words Based On Latin Number Names

  Results from FactBites:
 
Tuple relational calculus - Wikipedia, the free encyclopedia (2012 words)
The tuple calculus is a calculus that was introduced by Edgar F. Codd as part of the relational model in order to give a declarative database query language for this data model.
Along with the tuple calculus Codd also introduced the domain calculus which is closer to first-order logic and showed that these two calculi (and the relational algebra) are equivalent in expressive power.
The subset of C for which a tuple t is defined is called the domain of t (not to be confused with the domain in the schema) and denoted as dom(t).
Tuple - Wikipedia, the free encyclopedia (675 words)
In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects.
For example, a directed graph is defined as a tuple (V, E) where V is the set of nodes and E is a subset of V × V that denotes the edges.
In some languages and especially in database theory, a tuple is usually defined as a finite function that maps field names to a certain value.
  More results at FactBites »

 
 

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