# ordered tuplet

The concept of ordered $n$-tuplet is the generalization of ordered pair to $n$ items. For small values of $n$, the following 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 as a nested ordered pair looks like

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

This form of 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.

Title ordered tuplet OrderedTuplet 2013-03-22 14:55:44 2013-03-22 14:55:44 rspuzio (6075) rspuzio (6075) 16 rspuzio (6075) Definition msc 03-00 tuplet $n$-tuplet $n$-tuplets ordered $n$-tuplet -tuplet -tuplets tuple $n$-tuple ordered $n$-tupule -tuple finite sequence OrderedPair GeneralizedCartesianProduct triplet quadruplet quintuplet sextuplet septuplet octuplet nonuplet decuplet