Editing Tuple (section)

==Definitions==

There are several definitions of tuples that give them the properties described in the previous section.

===Tuples as functions===
The <math>0</math>-tuple may be identified as the [[Function (mathematics)#General properties|empty function]]. For <math>n \geq 1,</math> the <math>n</math>-tuple <math>\left(a_1, \ldots, a_n\right)</math> may be identified with the ([[Surjective function|surjective]]) [[Function (mathematics)#Definition|function]] 
:<math>F ~:~ \left\{ 1, \ldots, n \right\} ~\to~ \left\{ a_1, \ldots, a_n \right\}</math>

with [[Domain of a function|domain]] 
:<math>\operatorname{domain} F = \left\{ 1, \ldots, n \right\} = \left\{ i \in \N : 1 \leq i \leq n\right\}</math>

and with [[codomain]] 
:<math>\operatorname{codomain} F = \left\{ a_1, \ldots, a_n \right\},</math>

that is defined at <math>i \in \operatorname{domain} F = \left\{ 1, \ldots, n \right\}</math> by
:<math>F(i) := a_i.</math>

That is, <math>F</math> is the function defined by
:<math>\begin{alignat}{3}
1  \;&\mapsto&&\;  a_1 \\
   \;&\;\;\vdots&&\;      \\
n  \;&\mapsto&&\;  a_n \\
\end{alignat}</math>

in which case the equality 

:<math>\left(a_1, a_2, \dots, a_n\right) = \left(F(1), F(2), \dots, F(n)\right)</math>

necessarily holds.

;Tuples as sets of ordered pairs

Functions are commonly identified with their [[Graph of a function|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 <math>F</math> can be defined as:

:<math>F ~:=~ \left\{ \left(1, a_1\right), \ldots, \left(n, a_n\right) \right\}.</math>

===Tuples as nested ordered pairs===
Another way of modeling tuples in set theory is as nested [[ordered pair]]s. This approach assumes that the notion of ordered pair has already been defined.
# The 0-tuple (i.e. the empty tuple) is represented by the empty set <math>\emptyset</math>.
# An {{math|''n''}}-tuple, with {{math|''n'' > 0}}, can be defined as an ordered pair of its first entry and an {{math|(''n'' − 1)}}-tuple (which contains the remaining entries when {{math|''n'' > 1)}}:
#: <math>(a_1, a_2, a_3, \ldots, a_n) = (a_1, (a_2, a_3, \ldots, a_n))</math>
This definition can be applied recursively to the {{math|(''n'' − 1)}}-tuple:
: <math>(a_1, a_2, a_3, \ldots, a_n) = (a_1, (a_2, (a_3, (\ldots, (a_n, \emptyset)\ldots))))</math>

Thus, for example:
: <math>
    \begin{align}
         (1, 2, 3) & = (1, (2, (3, \emptyset)))      \\
      (1, 2, 3, 4) & = (1, (2, (3, (4, \emptyset)))) \\
    \end{align}
  </math>

A variant of this definition starts "peeling off" elements from the other end:
# The 0-tuple is the empty set <math>\emptyset</math>.
# For {{math|''n'' > 0}}:
#: <math>(a_1, a_2, a_3, \ldots, a_n) = ((a_1, a_2, a_3, \ldots, a_{n-1}), a_n)</math>
This definition can be applied recursively:
: <math>(a_1, a_2, a_3, \ldots, a_n) = ((\ldots(((\emptyset, a_1), a_2), a_3), \ldots), a_n)</math>

Thus, for example:
: <math>
    \begin{align}
         (1, 2, 3) & = (((\emptyset, 1), 2), 3)      \\
      (1, 2, 3, 4) & = ((((\emptyset, 1), 2), 3), 4) \\
    \end{align}
  </math>

===Tuples as nested sets===
Using [[ordered pair#Kuratowski's definition|Kuratowski's representation for an ordered pair]], the second definition above can be reformulated in terms of pure [[set theory]]:
# The 0-tuple (i.e. the empty tuple) is represented by the empty set <math>\emptyset</math>;
# Let <math>x</math> be an {{math|''n''}}-tuple <math>(a_1, a_2, \ldots, a_n)</math>, and let <math>x \rightarrow b \equiv (a_1, a_2, \ldots, a_n, b)</math>. Then, <math>x \rightarrow b \equiv \{\{x\}, \{x, b\}\}</math>.  (The right arrow, <math>\rightarrow</math>, could be read as "adjoined with".)

In this formulation:
: <math>
   \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}
  </math>