# Cartesian Products

Definition. Let S and T be sets. The Cartesian product of S and T is the set consisting of all ordered pairs , where and .

Ordered pairs are characterized by the following property: if and only if

Remarks. (a) is not the same as unless .

(b) You can define an ordered pair using sets. For example, the ordered pair can be defined as the set .

Example. Let and . List the elements of and sketch the set.

Notice that S and T are not subsets of . There are subset which "look like" S and T; for example, here's a subset that "looks like" S:

But this is not S: The elements of S are a, b, and c, whereas the elements of the subset U are pairs.

Here's a picture of . The elements are points in the grid:

consists of all pairs , where . This is the same thing as the the x-y-plane:

Example. Consider the following subset of :

(a) Prove that .

(b) Prove that .

(a)

(b) Suppose . Then for some , I have

Equating the first components, I get , so . But equating the second components, I get , so . This is a contradiction, so .

Example. is the set of pairs of integers. Consider the following subsets of :

Let . Then

You can take the product of more than 2 sets --- even an infinite number of sets, though I won't consider infinite products here.

For example, consists of ordered triples , where a, b, and c are integers.

Example. Consider the following subset of :

(a) Show that .

(b) Show that .

(a)

(b) Suppose . Then for some integers a and b, I have

Equating components, I get three equations:

But substituting and into gives