Help Find an Injective Function $f : P(\omega) \times P(\omega) \rightarrow P(\omega)$

Question

this is not a homework assignment but something I am curious to know before an exam in set theory coming up next week.

I suspect that the cardinality of $P(\omega) \times P(\omega)$ equals the cardinality of $P(\omega)$.

So I need to find one-to-one functions in each direction. Obviously one direction (right to left) is trivial. I'm having trouble finding or disproving the other direction (left to right).

I'm looking for an injective $f:P(\omega) \times P(\omega) \rightarrow P(\omega)$, or proof one doesn't exist.

Maybe you could point me in the right direction? Thanks

Answer 1

Hint: for a pair $(u,v)$ subsets of $\omega,$ map $u$ to the corresponding subset of the evens and $v$ to the corresponding subset of the odds.

Added: I think the map here is not only injective, but also surjective. To get an inverse of set $w$ we can take the evens in $w$ and arrive at an appropriate set $u,$ and similarly the odds in $w$ and arrive at a set $v,$ such that $(u,v)$ maps to $w$ by the above suggested map.