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A finite sequence (admitting repetitions) of elements from some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944301.png" />. A tuple is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944307.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944308.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944309.png" /> is called its length (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443010.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443011.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443012.png" />-th term of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443013.png" />-tuple and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443015.png" />). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443016.png" /> one finds the empty tuple, which contains no terms.

Synonyms of the term tuple are the following: a [[Word|word]] in the [[Alphabet|alphabet]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443017.png" /> (in this case it is usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443018.png" /> is finite); an element of some Cartesian power of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443019.png" />; an element of the free semi-group with identity generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443020.png" />; a function defined on the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443021.png" /> natural numbers (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443022.png" />) with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443023.png" />.



====Comments====
The typical property of tuples is that a tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443024.png" /> is equal to another one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443025.png" /> precisely when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443028.png" />. In the context of a set-theoretical foundation of mathematics (such as Zermelo–Fraenkel [[Set theory|set theory]]), where every object must be a set or a class, tuples are usually constructed as sets by the following inductive procedure: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443029.png" /> is the empty set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443030.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443031.png" />.