Anyone can assist me finding the cardinality of Complex Numbers and some of its subsets under ZFC?
and if we are to prove that if $\kappa$ is any uncountable cardinals, |$\omega \times \kappa$|=$\kappa$. Can it be proved by using if A $\subset$ B and if has uncountable cardinality then A has uncountable cardinality.
If you already know what the answer is you can check it using the Cantor–Bernstein–Schroeder theorem, which states that if $A$ and $B$ are sets and there are injective functions $f\colon A \to B$ and $g \colon B \to A$ then $A$ and $B$ have the same cardinality. So you should try and construct maps of this type for $A = \mathbb R$, $B = \mathbb R^2$. Note these do not have to be continuous or anything like that (although they might be, see the Peano curve). I suggest you construct these maps based on the decimal expansion, where the expansion of the output is determined by that of the input.
