3 Significance of tuples

A mathematical tuple is often represented as a linked list in data structure. As a result, all theories pertaining to tuples also apply to linked lists.

Perhaps even more importantly, tuples are important because they are the building blocks of ``functions''. In this context, a function is not a subroutine that returns a value. A function in this context is a mapping from a set of values to another set of values. A value in this context is represented by a tuple.

Tuples are also important because they can be used to represent the result of the Cartesian product of sets. The Cartesian product of two sets, $S$ and $T$ is defined as follows:

$S\times T = \{(e, f)\vert(e \in S) \wedge (f \in T)\}$

Here is an example:

$\{a, b, c\} \times \{x, y\} = \{ (a, x), (b, x), (c, x), (a, y), (b, y), (c, y)\}$

Note that the result of a Cartesian product is a set itself, which means it can be ``multiplied'' to another set. Furthermore, Cartesian products are associative, which means $(S \times T) \times U = S \times (T \times U)$ .