i recently started studying cardinal numbers and i got stuck on the first part of a question : 
i didnt know how to approach it i thought that since its an inverse function it has to be injective and surjective which makes it aleph null.. but i could not see how this will help me.
although the second part of it is easier , i needed to prove that ℵ1*ℵ1=ℵ1 according to the result from the first part which is easier (took A=natural even numbers , B= natural odd numbers so 2^|A|+|B| = ℵ1)
thank you all for the help on this site.
I take it that by inverse function your source means a bijection. Once you’ve shown that
$$f:\wp(A\cup B)\to\wp(A)\times\wp(B):X\mapsto\langle X\cap A,X\cap B\rangle$$
is a bijection, you know that $|\wp(A\cup B)|=|\wp(A)\times\wp(B)|$. To finish the job you need just a couple more facts. First, you need to know that $|\wp(S)|=2^{|S|}$ for any set $S$, and I expect that you do already have this. Secondly, you need to know that $|S\times T|=|S|\cdot|T|$ for any sets $S$ and $T$, and again I expect that you do already have this. Note that none of this has to do with $\aleph_0$ or any other particular cardinal number.
What you did for the second part is probably what you were expected to do, but if so, the person who composed the problem made a significant error: $2^{\aleph_0}=2^{|\Bbb N|}$ is not necessarily $\aleph_1$. $\aleph_1$ is by definition the smallest cardinal number that is larger than $\aleph_0$. The usual axioms of set theory simply do not determine whether $2^{\aleph_0}$ is $\aleph_1$ or some much larger cardinal number. The assertion that $2^{\aleph_0}=\aleph_1$ is known as the Continuum Hypothesis, and it has been known since 1963 that both it and its negation are consistent with the usual axioms of set theory.