nLab
tuple

Tuples

Idea

The generalisation of ordered pair to something having more positions is usually called a tuple (or ordered tuple). More particularly one gets the term nn-tuple, which refers to a list, (x 1,,x n)(x_1, \ldots, x_n), with nn entries from some set, XX; here nn is a natural number. It thus corresponds to an element in the nn-fold product set, X nX^n. The various elements x ix_i of the nn-tuple are usually called its components and sometimes it is useful to call the set of components the support or range of the tuple.

Variations

  • An ordered pair is a 22-tuple. A 33-tuple is a triple, a 44-tuple is a quadruple, a 55-tuple is a quintuple etc.

  • The notion of 11-tuple is trivial; a 11-tuple from XX is equivalent to an element of XX. But if you wish to emphasize that you have a 11-tuple, then it may be called a singleton.

  • The notion of 00-tuple is also trivial, but in a different way; there is (for each set XX) a single 00-tuple from XX. See empty list for notations, but in the end it hardly matters what you call it.

  • The term ‘tuple’ is usually used for an nn-tuple for a specific number nn. If we wish to speak of an nn-tuple for an arbitrary nn (particularly without specifying that nn), then we may speak of a list (which has other terminology, described on that page). Then the set of lists is the disjoint union over nn of the sets of nn-tuples.

  • The term ‘tuple’ is usually used for an nn-tuple for a finite number nn. If we wish to speak of an nn-tuple for an infinite (or possibly infinite) nn, then we may speak of a sequence.

Formalisation

See ordered pair for methods of formalising ordered pairs (which are 22-tuples) in various foundations of mathematics. Some of these generalise immediately to nn-tuples for arbitrary nn; otherwise, we may define nn-tuples recursively: a triple is an ordered pair whose (say) first component is an ordered pair; a quadruple is an ordered pair whose first component is a triple, etc.

Revised on April 6, 2017 00:15:39 by Toby Bartels (108.167.41.14)