## SectionB.3Cartesian 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 $$Y=\text{,}$$ 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 $$X\times Y=\emptyset\text{.}$$

### ExampleB.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*}