Tuple

The ${\displaystyle 0}${\displaystyle 0}-tuple may be identified as the empty function. For ${\displaystyle n\geq 1,}${\displaystyle n\geq 1,} the ${\displaystyle n}${\displaystyle n}-tuple ${\displaystyle \left(a_{1},\ldots ,a_{n}\right)}${\displaystyle \left(a_{1},\ldots ,a_{n}\right)} may be identified with the surjective function

${\displaystyle F~:~\left\{1,\ldots ,n\right\}~\to ~\left\{a_{1},\ldots ,a_{n}\right\}}${\displaystyle F~:~\left\{1,\ldots ,n\right\}~\to ~\left\{a_{1},\ldots ,a_{n}\right\}}

with domain

${\displaystyle \operatorname {domain} F=\left\{1,\ldots ,n\right\}=\left\{i\in \mathbb {N} :1\leq i\leq n\right\}}${\displaystyle \operatorname {domain} F=\left\{1,\ldots ,n\right\}=\left\{i\in \mathbb {N} :1\leq i\leq n\right\}}

and with codomain

${\displaystyle \operatorname {codomain} F=\left\{a_{1},\ldots ,a_{n}\right\},}${\displaystyle \operatorname {codomain} F=\left\{a_{1},\ldots ,a_{n}\right\},}

that is defined at ${\displaystyle i\in \operatorname {domain} F=\left\{1,\ldots ,n\right\}}${\displaystyle i\in \operatorname {domain} F=\left\{1,\ldots ,n\right\}} by

${\displaystyle F(i):=a_{i}.}${\displaystyle F(i):=a_{i}.}

That is, ${\displaystyle F}${\displaystyle F} is the function defined by

${\displaystyle {\begin{alignedat}{3}1\;&\mapsto &&\;a_{1}\\\;&\;\;\vdots &&\;\\n\;&\mapsto &&\;a_{n}\\\end{alignedat}}}${\displaystyle {\begin{alignedat}{3}1\;&\mapsto &&\;a_{1}\\\;&\;\;\vdots &&\;\\n\;&\mapsto &&\;a_{n}\\\end{alignedat}}}

in which case the equality

${\displaystyle \left(a_{1},a_{2},\dots ,a_{n}\right)=\left(F(1),F(2),\dots ,F(n)\right)}${\displaystyle \left(a_{1},a_{2},\dots ,a_{n}\right)=\left(F(1),F(2),\dots ,F(n)\right)}

necessarily holds.

Tuples as sets of ordered pairs

Functions are commonly identified with their graphs, which is a certain set of ordered pairs. Indeed, many authors use graphs as the definition of a function. Using this definition of "function", the above function ${\displaystyle F}${\displaystyle F} can be defined as:

${\displaystyle F~:=~\left\{\left(1,a_{1}\right),\ldots ,\left(n,a_{n}\right)\right\}.}${\displaystyle F~:=~\left\{\left(1,a_{1}\right),\ldots ,\left(n,a_{n}\right)\right\}.}

Another way of modeling tuples in set theory is as nested ordered pairs. This approach assumes that the notion of ordered pair has already been defined.

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set ${\displaystyle \emptyset }${\displaystyle \emptyset }.
2. An n-tuple, with n> 0, can be defined as an ordered pair of its first entry and an (n − 1)-tuple (which contains the remaining entries when n> 1):

   ${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},a_{3},\ldots ,a_{n}))}${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},a_{3},\ldots ,a_{n}))}

This definition can be applied recursively to the (n − 1)-tuple:

${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},(a_{3},(\ldots ,(a_{n},\emptyset )\ldots ))))}${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},(a_{3},(\ldots ,(a_{n},\emptyset )\ldots ))))}

Thus, for example:

${\displaystyle {\begin{aligned}(1,2,3)&=(1,(2,(3,\emptyset )))\\(1,2,3,4)&=(1,(2,(3,(4,\emptyset ))))\\\end{aligned}}}${\displaystyle {\begin{aligned}(1,2,3)&=(1,(2,(3,\emptyset )))\\(1,2,3,4)&=(1,(2,(3,(4,\emptyset ))))\\\end{aligned}}}

A variant of this definition starts "peeling off" elements from the other end:

1. The 0-tuple is the empty set ${\displaystyle \emptyset }${\displaystyle \emptyset }.
2. For n> 0:

   ${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=((a_{1},a_{2},a_{3},\ldots ,a_{n-1}),a_{n})}${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=((a_{1},a_{2},a_{3},\ldots ,a_{n-1}),a_{n})}

This definition can be applied recursively:

${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=((\ldots (((\emptyset ,a_{1}),a_{2}),a_{3}),\ldots ),a_{n})}${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=((\ldots (((\emptyset ,a_{1}),a_{2}),a_{3}),\ldots ),a_{n})}

Thus, for example:

${\displaystyle {\begin{aligned}(1,2,3)&=(((\emptyset ,1),2),3)\\(1,2,3,4)&=((((\emptyset ,1),2),3),4)\\\end{aligned}}}${\displaystyle {\begin{aligned}(1,2,3)&=(((\emptyset ,1),2),3)\\(1,2,3,4)&=((((\emptyset ,1),2),3),4)\\\end{aligned}}}

Using Kuratowski's representation for an ordered pair, the second definition above can be reformulated in terms of pure set theory:

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set ${\displaystyle \emptyset }${\displaystyle \emptyset };
2. Let ${\displaystyle x}${\displaystyle x} be an n-tuple ${\displaystyle (a_{1},a_{2},\ldots ,a_{n})}${\displaystyle (a_{1},a_{2},\ldots ,a_{n})}, and let ${\displaystyle x\rightarrow b\equiv (a_{1},a_{2},\ldots ,a_{n},b)}${\displaystyle x\rightarrow b\equiv (a_{1},a_{2},\ldots ,a_{n},b)}. Then, ${\displaystyle x\rightarrow b\equiv \{\{x\},\{x,b\}\}}${\displaystyle x\rightarrow b\equiv \{\{x\},\{x,b\}\}}. (The right arrow, ${\displaystyle \rightarrow }${\displaystyle \rightarrow }, could be read as "adjoined with".)

In this formulation:

${\displaystyle {\begin{array}{lclcl}()&&&=&\emptyset \\&&&&\\(1)&=&()\rightarrow 1&=&\{\{()\},\{(),1\}\}\\&&&=&\{\{\emptyset \},\{\emptyset ,1\}\}\\&&&&\\(1,2)&=&(1)\rightarrow 2&=&\{\{(1)\},\{(1),2\}\}\\&&&=&\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\}\\&&&&\\(1,2,3)&=&(1,2)\rightarrow 3&=&\{\{(1,2)\},\{(1,2),3\}\}\\&&&=&\{\{\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\}\},\\&&&&\{\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\},3\}\}\\\end{array}}}${\displaystyle {\begin{array}{lclcl}()&&&=&\emptyset \\&&&&\\(1)&=&()\rightarrow 1&=&\{\{()\},\{(),1\}\}\\&&&=&\{\{\emptyset \},\{\emptyset ,1\}\}\\&&&&\\(1,2)&=&(1)\rightarrow 2&=&\{\{(1)\},\{(1),2\}\}\\&&&=&\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\}\\&&&&\\(1,2,3)&=&(1,2)\rightarrow 3&=&\{\{(1,2)\},\{(1,2),3\}\}\\&&&=&\{\{\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\}\},\\&&&&\{\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\},3\}\}\\\end{array}}}