* 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

This contradiction proves that .

Copyright 2018 by Bruce Ikenaga