Editing Tuple (section)

== Type theory ==
{{main|Product type}}

In [[type theory]], commonly used in [[programming language]]s, a tuple has a [[product type]]; this fixes not only the length, but also the underlying types of each component. Formally:
: <math>(x_1, x_2, \ldots, x_n) : \mathsf{T}_1 \times \mathsf{T}_2 \times \ldots \times \mathsf{T}_n</math>
and the [[Projection (mathematics)|projection]]s are term constructors:
: <math>\pi_1(x) : \mathsf{T}_1,~\pi_2(x) : \mathsf{T}_2,~\ldots,~\pi_n(x) : \mathsf{T}_n</math>

The tuple with labeled elements used in the [[relational model]] has a [[Record (computer science)|record type]]. Both of these types can be defined as simple extensions of the [[simply typed lambda calculus]].<ref name="pierce2002">{{cite book|last=Pierce|first=Benjamin|title=Types and Programming Languages|url=https://archive.org/details/typesprogramming00pier_207|url-access=limited|publisher=MIT Press|year=2002|isbn=0-262-16209-1|pages=[https://archive.org/details/typesprogramming00pier_207/page/n149 126]–132}}</ref>

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 theory|model]] of a type theory, and use the Scott brackets to indicate the semantic interpretation<!-- do not link; that article needs to be a dab first-->, then the model consists of some sets <math>S_1, S_2, \ldots, S_n</math> (note: the use of italics here that distinguishes sets from types) such that:
: <math>[\![\mathsf{T}_1]\!] = S_1,~[\![\mathsf{T}_2]\!] = S_2,~\ldots,~[\![\mathsf{T}_n]\!] = S_n</math>
and the interpretation of the basic terms is:
: <math>[\![x_1]\!] \in [\![\mathsf{T}_1]\!],~[\![x_2]\!] \in [\![\mathsf{T}_2]\!],~\ldots,~[\![x_n]\!] \in [\![\mathsf{T}_n]\!]</math>.

The {{math|''n''}}-tuple of type theory has the natural interpretation as an {{math|''n''}}-tuple of set theory:<ref>Steve Awodey, [http://www.andrew.cmu.edu/user/awodey/preprints/stcsFinal.pdf ''From sets, to types, to categories, to sets''], 2009, [[preprint]]</ref>
: <math>[\![(x_1, x_2, \ldots, x_n)]\!] = (\,[\![x_1]\!], [\![x_2]\!], \ldots, [\![x_n]\!]\,)</math>
The [[unit type]] has as semantic interpretation the 0-tuple.