Tuple
A finite sequence (admitting repetitions) of elements from some set . A tuple is denoted by
,
,
,
,
,
, or
. The number
is called its length (
),
is called the
-th term of the
-tuple and
(
). For
one finds the empty tuple, which contains no terms.
Synonyms of the term tuple are the following: a word in the alphabet (in this case it is usually assumed that
is finite); an element of some Cartesian power of the set
; an element of the free semi-group with identity generated by
; a function defined on the first
natural numbers (
) with values in
.
Comments
The typical property of tuples is that a tuple is equal to another one
precisely when
and
for all
. In the context of a set-theoretical foundation of mathematics (such as Zermelo–Fraenkel set theory), where every object must be a set or a class, tuples are usually constructed as sets by the following inductive procedure:
is the empty set for
, and
.
Tuple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tuple&oldid=31808