what is the cardinality of powerset of a union set?

Question

Is there exist something like P(X+Y) (P STANDS FOR POWERSET)? I am confuse because power set is the set of all subset of Cartesian product, and X+Y wont give Cartesian product but (x,0) U (y,1), and if it exist what is the cardinality of that power set ?Thank you.

Answer 1

Actually the power set is the set of all subsets: $$\mathcal P(X) = \{ M | M\subset X\}$$ So $\mathcal P(X\cup Y) = \{ M | M\subset X\cup Y\}$. If we assume $X\cap Y = \emptyset$ we can write this as $$\mathcal P(X\cup Y) = \{ M_X \cup M_Y | M_X\subset X, M_Y \subset Y\} = \mathcal P(X) \bigcup \mathcal P(Y) \simeq \mathcal P(X) \times \mathcal P(Y)$$ Where $\bigcup$ is defined as the set of unions of all elements. The last isomorphism gives us the cardinality for the disjoint case: $$\mathrm{card}(\mathcal P(X\cup Y)) = \mathrm{card}(\mathcal P(X)) \cdot \mathrm{card}(\mathcal P(Y))$$

Answer 2

If $|X+Y|=|X|+|Y|$ ,

then $|\mathcal{P}(X+Y)|=2^{|X+Y|}=2^{|X|+|Y|}=2^{|X|}2^{|Y|}= |\mathcal{P}(X)| |\mathcal{P}(Y)|$