Section 0.3 Tuples and Cartesian products
We now introduce the notion of a tuple, which is a formal description of an ordered collection of objects.
Definition 0.3.1. \(n\)-tuple.
Let \(A\) be a set, and fix a positive integer \(n\text{.}\) An \(n\)-tuple (or sequence of length \(n\)) of elements of \(A\) is an ordered sequence \((a_1,a_2,\dots, a_n)\) where \(a_i\in A\) for all \(1\leq i\leq n\text{.}\)
We define two \(n\)-tuples \((a_1,a_2,\dots, a_n)\text{,}\) and \((a_1',a_2',\dots, a_n')\) to be equal, denoted \((a_1,a_2,\dots, a_n)=(a_1',a_2',\dots, a_n')\text{,}\) if \(a_i=a_i'\) for all \(1\leq i\leq n\text{.}\)
We call \(n\) the length of the tuple \((a_1,a_2,\dots, a_n)\text{,}\) and for each \(1\leq i\leq n\) we call \(a_i\) its \(i\)-th entry or coordinate.
Definition 0.3.4. Cartesian product (finite).
Let \(A_1, A_2, \dots, A_n\) be subsets of a common set \(A\text{.}\) The (Cartesian) product of \(A_1, A_2,\dots, A_n\text{,}\) denoted \(A_1\times A_2\times\cdots \times A_n\) or \(\displaystyle\prod_{i=1}^nA_i\text{,}\) is the set
\begin{equation*}
\prod_{i=1}^nA_i=\{(a_1,a_2,\dots, a_n)\colon a_i\in A_i \text{ for all } 1\leq i\leq n\}\text{.}
\end{equation*}
In other words \(\prod_{i=1}^nA_i\) is the set of all \(n\)-tuples of \(A\) whose \(i\)-th coordinate lies in \(A_i\) for all \(1\leq i\leq n\text{.}\)
Given a set \(A\text{,}\) its \(n\)-ary Cartesian product \(A^n\) is defined as
\begin{equation*}
A^n=\prod_{i=1}^n A=\underset{n\text{ times}}{\underbrace{A\times A\times\cdots \times A}}\text{.}
\end{equation*}
The notion of Cartesian product can be generalized to an infinite list of sets \(A_1, A_2, \dots\text{,}\) and indeed to any collection \(\{A_i\}_{i\in I}\) indexed by a set \(I\text{.}\) This is accomplished by looking at tuples in a slightly different manner: namely, we can describe a tuple \((a_1,a_2,\dots, a_n)\in \prod_{k=1}^nA_i\) as an assignment to each distinct element \(i\in \{1,2,\dots, n\}\) an element \(a_i\in A_i\) that we call the coordinate of \(i\text{.}\) In other words, a tuple \((a_1,a_2,\dots, a_n)\) is just a function that assigns to each \(i\) in our index set \(\{1,2,\dots, n\}\) an element \(a_i\in A_i\text{.}\) This notion generalize easily by replacing the finite index set \(\{1,2,\dots, n\}\) with an arbitrary set \(I\) (finite or infinite).
Definition 0.3.5. I-tuple.
Let \(I\) be a set. Given a set \(A\text{,}\) an \(I\)-tuple of elements of \(A\) is a function \(f\colon I\rightarrow A
\text{.}\) Given an \(I\)-tuple \(f\) and element \(i\in I\) we will often denote the value \(f(i)\) as \(a_i\text{,}\) and denote \(f\) itself as \(f=(a_i)_{i\in I}\text{.}\) In analogy to finite tuples, we call \(a_i\) the \(i\)-th entry or coordinate of \(f\text{.}\)
Definition 0.3.6. Cartesian product (arbitrary).
Let \(A\) be a set, and let \(\{A_i\}_{i\in I}\) be a collection of subsets \(A_i\subseteq A\) indexed by the set \(I\text{.}\) The Cartesian product \(\prod_{i\in I}A_i\) of this collection is defined as
\begin{align*}
\prod_{i\in I}A_i\amp =\{f\colon I\rightarrow A\colon f(i)\in A_i \text{ for all } i\in I\} \\
\amp =\{(a_i)_{i\in I}\colon a_i\in A_i \text{ for all } i\in I\}\text{.}
\end{align*}
In other words, the Cartesian product is the set of all \(I\)-tuples of elements of \(A\) whose \(i\)-th coordinate is an element of \(A_i\) for all \(i\in I\text{.}\)
In the special case where \(A_i=A\) for all \(i\in I\text{,}\) we denote \(\prod_{i\in I}A\) as \(A^I\text{.}\)
