Question

Suppose that \(|A|=m\) and \(|B|=n .\) Find the following cardinalities. $$ |\mathscr{P}(A) \times \mathscr{P}(B)| $$

Short answer

The cardinality of the Cartesian product of the power sets of A and B is \(2^{m+n}\)

Step-by-step solution

Compute Power Sets Cardinalities

Firstly, calculate the cardinalities of the power sets of A and B. This is done by using the relation \(|\mathscr{P}(X)| = 2^{|X|}\), where \(|X|\) is the cardinality of set X. Therefore, \(|\mathscr{P}(A)| = 2^m\) and \(|\mathscr{P}(B)| = 2^n\).

Compute Cartesian Product Cardinality

Next, calculate the cardinality of the Cartesian product of the power sets \(\mathscr{P}(A)\) and \(\mathscr{P}(B)\). The cardinality of the Cartesian product of two sets is the product of their cardinalities. Thus, \(|\mathscr{P}(A) \times \mathscr{P}(B)| = |\mathscr{P}(A)| \cdot |\mathscr{P}(B)|\).

Substitution

Substitute our values from step 1 into the formula from step 2 to get the final answer, \(|\mathscr{P}(A) \times \mathscr{P}(B)| = 2^m \cdot 2^n = 2^{m+n}\)