Question

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

Short answer

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

Step-by-step solution

Determine the cardinality of the Cartesian product

The Cartesian Product of two sets A and B is the set of all ordered pairs where the first element is from set A and the second element is from set B. The cardinality of a Cartesian Product |A x B| is given by \(|A|\) x \(|B|\). Since it is given that \(|A|=m\) and \(|B|=n\), the cardinality of \(A x B\) will be given by m x n.

Find the cardinality of the power set

The power set of a set X is the set of all its possible subsets. The cardinality of a power set \(|\mathscr{P}(X)|\) is given by \(2^{|X|}\), where \(|X|\) is the cardinality of the original set. Applying this formula to the result from Step 1, the cardinality of \(|\mathscr{P}(A x B)|\) is \(2^{m x n}\).

Final Answer

The cardinality of the power set \(|\mathscr{P}(A x B)|\) is \(2^{m x n}\).