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.
We have more suggestive names for \(n\)-tuples when \(n\) is small: a 2-tuple \((a,b)\) is called a pair, a 3-tuple \((a,b,c)\) is called a triple, a 4-tuple \((a,b,c,d)\) is called a quadruple, etc.. We will use the generic term “tuple” to designate a \(n\)-tuple of unspecified length.
Observe how tuples capture the notion of an ordered collection of objects. For example, whereas the sets \(\{3, 1, 2, 4\}\) and \(\{1,2,3,4\}\) are equal to one another, the 4-tuples \((3,1,2,4)\) and \((1,2,3,4)\) are not.
What about the tuples \((1,1,1)\) and \((1,1,1,1)\text{?}\) Are these equal? Technically our definition of equality only applies to tuples living in the same fixed Cartesian product. In particular, for the question of equality to make sense, the tuples must have the same length. As such we will officially avoid writing things like \((1,1,1)\ne (1,1,1,1)\text{,}\) although unofficially we consider these two objects as completely different. You should think of \((1,1,1)\) and \((1,1,1,1)\) as creatures living on two different planets in the universe of tuples.
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 \(1\leq i\leq n\text{.}\)
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).
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 \(f\text{.}\)
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 \(i\in I\text{.}\)
