# The Cartesian Product of Sets

Definition: Let $A$ and $B$ be sets. Then the Cartesian Product of $A$ and $B$ denoted $A \times B$ is defined to be the set of ordered pairs $(a, b)$ where $x \in A$ and $y \in B$, that is $A \times B = \{ (x, y) : x \in A \: \mathrm{and} \: y \in B \}$. |

*Note that the definition of $A \times B$ requires that each pair of elements is ordered in the sense that for each $(x, y) \in A \times B$ we MUST have that $x \in A$ and $y \in B$.*

For example, consider the following sets:

(1)Then we have that:

(3)Note that the first element in each ordered pair is an element of $A$ and the second element in each ordered pair is an element of $B$. The pair $(x, a) \not \in A \times B$ though since $x \not \in A$ AND $a \not \in B$. While the order of elements in a set does not matter, the order of a list - or in this case, the order of our pair or any $n$-tuple is indeed important.

For a more geometric interpretation for the Cartesian product of two sets of real numbers, consider the following sets:

(4)If we regard each ordered pair $(x, y)$ as a point on the Cartesian coordinate system, $\mathbb{R}^2$, then we have that:

(6)Geometrically this can be represented with the following illustration:

Of course, we can construct the Cartesian product of more than two sets in a similar manner. Let $A_1, A_2, ..., A_n$ be a collection of sets. Then the Cartesian product of these sets with this prescribed order as the set of *ordered* $n$-tuples: