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
:
Prove that .
Let . B consists of pairs whose
components add to an odd number. So I add the components of
:
Since is odd,
is odd. This proves that
.
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 2019 by Bruce Ikenaga