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# Prove that the “number of elements” set-function satisfies Axioms 1-3 of area

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, 