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Section B.3 Cartesian Products

When \(X\) and \(Y\) are sets, the cartesian product of \(X\) and \(Y\text{,}\) denoted \(X\times Y\), is defined by

\begin{equation*} X\times Y=\{(x,y): x\in X \text{ and } y\in Y\} \end{equation*}

For example, if \(X=\{a,b\}\) and \(Y=[3]\text{,}\) then

\begin{equation*} X\times Y=\{(a,1),(b,1),(a,2),(b,2),(a,3),(b,3)\}. \end{equation*}

Elements of \(X\times Y\) are called ordered pairs. When \(p=(x,y)\) is an ordered pair, the element \(x\) is referred to as the first coordinate of \(p\) while \(y\) is the second coordinate of \(p\text{.}\) Note that if either \(X\) or \(Y\) is the empty set, then \(X\times Y=\emptyset\text{.}\)

Example B.3.

Let \(X=\{\emptyset,(1,0),\{\emptyset\}\}\) and \(Y=\{(\emptyset,0)\}\text{.}\) Is \(((1,0),\emptyset)\) a member of \(X\times Y\text{?}\)

Cartesian products can be defined for more than two factors. When \(n\ge 2\) is a positive integer and \(X_1,X_2,\dots,X_n\) are non-empty sets, their cartesian product is defined by

\begin{equation*} X_1\times X_2\times\dots\times X_n=\{(x_1,x_2,\dots,x_n): x_i\in X_i \text{ for } i = 1,2,\dots,n\} \end{equation*}