Editing Tuple (section)

===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>