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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. Wikipedia does not have an article with this exact name. ...
Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ...
Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
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. ...
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 Look up zero in Wiktionary, the free dictionary. ...
Look up one in Wiktionary, the free dictionary. ...
Generally, a singleton is something which exists alone in some way. ...
Look up double in Wiktionary, the free dictionary. ...
This article does not cite any references or sources. ...
The word pair, derived via the French words pair/paire from the Latin par equal, can mean: the French equivalent of peer, holder of a French Pairie, French high title roughly equivalent to a member of the British peerage 2 (number), two of something Couple, various senses for two joined...
2 (two) is the natural number following 1 and preceding 3. ...
In mathematics, a triple is an n-tuple with n being 3. ...
This article discusses the number three. ...
A triplet is a set of three items, and includes in particular: one of three babies in a multiple birth in lapidary, a preparation of opal as a gemstone, with a thin layer of opal backed with a dark material and covered with cap of clear quartz in optics a...
Treble is a term applied in music to the high or acute part of the musical system, as opposed to the bass, the lower or grave part. ...
This article is about the post-hardcore band. ...
For the 1994 film, see Threesome (film). ...
A general meaning of the Russian word troika (Cyrillic alphabet: тройка) is threesome, a collection of three of any kind. ...
Look up trio in Wiktionary, the free dictionary. ...
A quadruple is a term from mathematics, depicting an n-tuple with n being 4. ...
This article discusses the number Four. ...
Look up five in Wiktionary, the free dictionary. ...
Look up six in Wiktionary, the free dictionary. ...
Seven Days of Creation - 1765 book, title page 7 (seven) is the natural number following 6 and preceding 8. ...
Look up eight in Wiktionary, the free dictionary. ...
Look up nine in Wiktionary, the free dictionary. ...
10 (ten) is an even natural number following 9 and preceding 11. ...
100 (one hundred) (the Roman numeral is C for centum) is the natural number following 99 and preceding 101. ...
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 objects considered as a whole. ...
- it can contain an object more than once, and
- 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 ℕ: Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
- (a1, a2, …,an) ≡ fa: ℕn → A: i ↦ ai.
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). ...
- the 0-tuple (i.e. the empty tuple) is represented by ∅
- 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. ...
- a special symbol NIL represents the empty list;
- 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 has 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 ISO-standardized object-oriented programming language designed for extensibility, reusability, reliability and programmer productivity. ...
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.
Information Modeling With inherant Name/Value pair properties, along with a structured and ordered nature, the term 'tuple' extends readily to use in Information Modeling and Database Definition.
For example, XML-Tuples represent Name/Value tuple structures. The following is an example of an XML-Tuple
<name>Value</name>
Field names 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. ...
- ( value : 16, previous-node : 1174782, next-node : 1174791 )
See also In mathematics, the Cartesian product is a direct product of sets. ...
In mathematics, logic, and computer science, a formal language is a language that is defined by precise mathematical or machine processable formulas. ...
Eiffel is an ISO-standardized object-oriented programming language designed for extensibility, reusability, reliability and programmer productivity. ...
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, the concept of a relation is a generalization of 2-place relations, such as the relation of equality, denoted by the sign = in a statement like 5 + 7 = 12, or the relation of order, denoted by the sign < in a statement like 5 < 12. Relations that involve two...
This article may be too technical for most readers to understand. ...
A unit type is a mathematical type that allows only one value. ...
In logic, mathematics, and computer science, the arity (synonyms include type, adicity, and rank) of a function or operation is the number of arguments or operands that the function takes. ...
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