Let
where is power set of
(i.e., the class of all subsets of
). Next, define a function
by
equals the number of distinct elements in
for any
. Then, let
Compute
Then, prove the function satisfies Axioms 1-3 for area.
First, we compute
Next, we prove this satisfies the first three area axioms.
Proof.
Axiom 1. (Non-negative property) This is satisfied for any set since the number of distinct elements in a set is non-negative. So,
for all
.
Axiom 2. (Additive property) First, if , then
by definition of
. So, for any
we have
and for any
, we have
.
Thus, if , then
; hence,
, so
.
Then, implies
(since
). Hence,
.
So, for any we have
.
Next, we must show . For any
we have
, or
and
. So, this means
, or
or
. Thus,
Similarly, we note,
So,
Axiom 3. (Difference property) If and
, then from above we have
But, for we know
, so,