Tuples#
Definition: Tuple
The \(0\)-tuple is the empty set \(\varnothing\):
\[() \overset{\text{def}}{=} \varnothing\]
A \(1\)-tuple \((x)\) is just \(x\) itself:
\[(x) \overset{\text{def}}{=} x\]
An \(n\)-tuple is defined recursively via sets in the following way:
\[(x_1,\dotsc,x_n) \overset{\text{def}}{=} \begin{cases}\{\{x_1\},\{x_1,x_2\}\} & \text{if } n = 2 \\ \\ ((x_1, \dotsc, x_{n-1}), x_n) & \text{if } n \gt 2\end{cases}\]
Definition: Ordered Pair
The ordered pair \((a;b)\) of two objects \(a\) and \(b\) is the collection
\[ (a;b) \overset{\text{def}}{=} \{\{a\},\{a,b\}\} \]
Theorem: Equality of Ordered Pairs
Two ordered pairs \((a;b)\) and \((c;d)\) are equal if and only if \(a = b\) and \(c = d\).
\[ (a;b) = (c;d) \iff (a = b \land c=d) \]
Proof
TODO
Note
This property of ordered pairs means that, in general, \((a;b) \ne (b;a)\), hence the name "ordered".
Tuples#
Definition: \(n\)-Tuple
An \(n\)-tuple \((a_1,\cdots,a_n)\) is a collection of \(n\) ordered pairs \(\{(1,a_1), \cdots, (n,a_n)\}\).