Power sets allow us to discuss the class of all subsets of a given set
, i.e.
. That this is a set is the subject of the Power Set Axiom.
Axiom
Given a set
there exists a set of sets
such that
iff
.
Theorem Given a set
, there exists a unique set whose elements are the subsets of
.
Proof If
and
are two such sets of subsets then
if and only if
. But the same is true of
. Thus
iff
, and so
by the Axiom of Extensionality.
Definition Given a set
, the set of all subsets of
is called the power set of
. It is denoted
.
Example If
then
.
Recall the Kuratowski definition of an ordered pair,
for
and
elements of a set
. Note that
and
are both subsets of
, i.e. they are elements of the power set
.
This means that
is a subset of
, i.e.
.
We can generalise this slightly with a simple trick. We can define
with
and
for sets
and
. In order to do this, we simply take the elements
and
from the union of sets
.
In other words, we have
with
and
.
Theorem The class of all ordered pairs
of elements of
with
and
, is a set.
Proof The set in question is given by
. This is a set by the axioms of Power Set, Union and the Axiom Schema of Comprehension.
Definition The set of ordered pairs
with
and
is called the cartesian product of
and
, and is denoted
.
- Show that for sets
we have
.
- Show that for sets
we have
.
- Show that for sets
we have
.
- Show that for sets
with
we have
.
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