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The n can be replaced by a specific number, thus one can for example say a quaternion can be represented as a 4tuple. A 2tuple is an ordered pair; a 3tuple is a triple or triplet; further constructions are possible, such as octuple, but many mathematicians find it quicker to write "8tuple", even if still pronouncing "octuple".
A general ntuple is: (a_{1},a_{2},...,a_{n})= (b_{1},b_{2},...,b_{n}) iff a_{1}=b_{1}, a_{2}=b_{2} and so on.
Formally, an ntuple can be defined in terms of sets as either (a_{1},a_{2},...,a_{n})= {a_{1},{a_{1},a_{2}},{a_{1},a_{2},a_{3}},...,{a_{1},a_{2},...a_{n}}} or as an inductive definition:
It is fairly easy to show that either definition implies the property given above. However, the sets generated look very different.
Many computer programming languages support tuples as a data type, either for objects of fixed types, or as a collection of objects of any type.
The Lisp programming language originally used the ordered pair abstraction to create all of its ntuple and list structures, similarly to the inductive definition above.
In the field of relational databases, a tuple is a row in a relation (table). An ntuple is a row with n columns. It is important not to confuse an ntuple with n tuples.
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