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 $n$-tuple, which refers to a list, $(x_1, \ldots, x_n)$, with $n$ entries from some set, $X$; here $n$ is a natural number. It thus corresponds to an element in the $n$-fold product set, $X^n$. The various elements $x_i$ of the $n$-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 $2$-tuple. A $3$-tuple is a triple, a $4$-tuple is a quadruple, a $5$-tuple is a quintuple etc.

• The notion of $1$-tuple is trivial; a $1$-tuple from $X$ is equivalent to an element of $X$. But if you wish to emphasize that you have a $1$-tuple, then it may be called a singleton.

• The notion of $0$-tuple is also trivial, but in a different way; there is (for each set $X$) a single $0$-tuple from $X$. See empty list for notations, but in the end it hardly matters what you call it.

• The term ‘tuple’ is usually used for an $n$-tuple for a specific number $n$. If we wish to speak of an $n$-tuple for an arbitrary $n$ (particularly without specifying that $n$), 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 $n$ of the sets of $n$-tuples.

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

Formalisation

See ordered pair for methods of formalising ordered pairs (which are $2$-tuples) in various foundations of mathematics. Some of these generalise immediately to $n$-tuples for arbitrary $n$; otherwise, we may define $n$-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.

Related concepts

• A 2-tuple is a pair.

• A 3-tuple is triple.

• tuple, sequence, function

• dependent tuple, dependent sequence, dependent function

• list, indexed set