This question has been suggested to me by this post : How to prove that $|P(A)| + |P(B)| = 16$?
Suppose A and B are disjoint.
If I want to add the cardinal of P(A) and of P(B) , I cannot use this formula
#X + #Y = # ( X U Y )
for here my two sets are not disjoint ( two powersets always having at least one element in common).
Here I know that the sets P(A) and P(B) only have one element in common ( since A and B are disjoint).
Is there a formula of cardinal arithmetics that applies here?
For any sets $X$ and $Y$, $|X\cup Y| = |X| + |Y| - |X\cap Y|$.
That is, you count up all the elements assuming they're all different, then you subtract the elements they have in common so that you don't double-count.
In particular, the power sets $P(A)$ and $P(B)$ of disjoint sets have one element in common—the empty set. So, count up the number of elements in each power set assuming they have nothing in common, then subtract the one element $(\varnothing)$ they have in common:
$$| P(A) \cup P(B)| = |P(A)| + |P(B)| - 1 = 2^{|A|} + 2^{|B|}-1$$