Definition: Cartesian Product

Defintion: Let A and B be sets. The Cartesian Product of A and B, which is often denoted by $A \times B$ , is the set of pairs of elements $(a, b)$ such that $a\in A$ and $b\in B$ . In symbols, we have $A\times B=\{(a,b)|a\in A \mathrm{\ and\ } b\in B\}$ .//

First off, if the notation in the last line of the definition is unfamiliar, refer to about half-way down lesson 16 for a reminder of what that all means. The important thing to note about this definition is that we have created new elements from the elements that A and B already contained. We can think of these new elements as “ways of picking exactly one element from A and one element from B”, so that it doesn’t really matter how we write these elements down. In other words, if we wrote the elements as $[[[a]]][[[b]]]$ or as $\{(a, b>]$ , or in any other crazy way, then we’d obviously have, in some sense, the same information as we do when we write them as $(a,b)$ . For more on this construction, take a look at lesson 17 where we explain this definition a lot more thoroughly.