Cartesian product of Power Sets

Question

I have to prove or disprove the following:

$$\mathcal{P}(A \times B)= \mathcal{P}(A) \times \mathcal{P}(B)$$

where $\mathcal{P}$ here is power set. I tried to do it using the definition, but I feel like there should be an easier way of proving or disproving it.

Does anyone have any pointers?

Answer 1

One way to see that it cannot always be true is to examine the cardinalities of the two sets.

There are $2^{|A|}$ elements in the powerset of some set $A$. The cardinality of the cartesian product $A \times B$ of two sets is $|A||B|$.

Using the above information, we have $$|\mathcal{P}(A \times B)|=2^{|A \times B|}=2^{|A||B|}$$ and $$|\mathcal{P}(A) \times \mathcal{P}(B)|=|\mathcal{P}(A)||\mathcal{P}(B)|=2^{|A|}2^{|B|}=2^{|A|+|B|}.$$

So we have shown that $$\mathcal{P}(A \times B)= \mathcal{P}(A) \times \mathcal{P}(B)$$

is not true when $|A||B|\neq|A|+|B|$.

Answer 2

It is false for all sets $A,B$. We have $\emptyset \in \mathcal{P}(A\times B)$ but $\emptyset \notin (\mathcal{P}(A)\times \mathcal{P}(B))$.