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,