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# ordered tuplet

The concept of ordered $n$-tuplet is the generalization of ordered pair to $n$ items. For small values of $n$, the following terms are used:

$\begin{matrix}n&\hbox{\sl name}\hfill&\hbox{\sl example}\hfill\\ 3&\hbox{triplet}\hfill&(a,b,c)\hfill\\ 4&\hbox{quadruplet}\hfill&(a,b,c,d)\hfill\\ 5&\hbox{quintuplet}\hfill&(a,b,c,d,e)\hfill\\ 6&\hbox{sextuplet}\hfill&(a,b,c,d,e,f)\hfill\\ 7&\hbox{septuplet}\hfill&(a,b,c,d,e,f,g)\hfill\\ 8&\hbox{octuplet}\hfill&(a,b,c,d,e,f,g,h)\hfill\\ 9&\hbox{nonuplet}\hfill&(a,b,c,d,e,f,g,h,i)\hfill\\ 10&\hbox{decuplet}\hfill&(a,b,c,d,e,f,g,h,i,j)\hfill\\ \end{matrix}$ |

This notion can be defined set-theoretically in a number of ways. For convenience, we shall express two of these definitions for quintuplets — it is perfectly easy to generalize them to any other value of $n$.

One possibility is to build $n$-tuplets out of nested ordered pairs. In the case of our example $(a,b,c,d,e)$, the representation as a nested ordered pair looks like

$(a,(b,(c,(d,e)))).$ |

This form of representation is used in the programming language LISP.

Another possibility is to define $n$-tuplets as maps. In this way of thinking, a quintuplet is a function whose domain is the set $\{1,2,3,4,5\}$. In the case of our example, the function $f$ in question is defined as

$\begin{array}[]{ccc}f(1)&=&a\\ f(2)&=&b\\ f(3)&=&c\\ f(4)&=&d\\ f(5)&=&e\\ \end{array}$ |

Especially with the second interpretation, one sees that a synonym for ”ordered tuplet” is ”finite sequence” or ”list”. For instance, a quintuplet can also be regarded as a sequence of five items or a list of five items.

## Mathematics Subject Classification

03-00*no label found*

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