Tuple
Type theory

In type theory, commonly used in programming languages, a tuple has a product type; this fixes not only the length, but also the underlying types of each component. Formally:

and the projections are term constructors:

The tuple with labeled elements used in the relational model has a record type. Both of these types can be defined as simple extensions of the simply typed lambda calculus.

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 of a type theory, and use the Scott brackets to indicate the semantic interpretation, then the model consists of some sets S 1, S 2, …, S n {\displaystyle S_{1},S_{2},\ldots,S_{n}} (note: the use of italics here that distinguishes sets from types) such that:

and the interpretation of the basic terms is:

The n -tuple of type theory has the natural interpretation as an n -tuple of set theory:

The unit type has as semantic interpretation the 0-tuple.
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