Question

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

Short answer

The cardinality of the given set is 2.

Step-by-step solution

Simplify A x ∅

The first step is to simplify \(A \times \varnothing\). The Cartesian product of any set with the empty set is the empty set. Therefore, \(A \times \varnothing = \varnothing\).

Simplify the above output within ∅

The next step is to work out Power set of \( \mathscr{P}(\varnothing)\). The power set of an empty set is a set that contains the empty set, therefore \( \mathscr{P}(\varnothing) = \{ \varnothing \}\).

Simplify ∅ within power set

The third step is to find the power set of \(\mathscr{P}(\{\varnothing\})\). The power set of a set that only contains the empty set is a set that contains the empty set and the set itself, therefore \(\mathscr{P}(\{\varnothing\}) = \{\varnothing, \{\varnothing\}\}\).

Find the cardinality

Lastly, calculate the cardinality of the final power set \(\mathscr{P}(\{\varnothing, \{\varnothing\}\})\). The cardinality of a set is the number of elements in the set. The set \(\{\varnothing, \{\varnothing\}\}\) has two elements, so its cardinality is 2. Therefore, \(|\mathscr{P}(\mathscr{P}(\mathscr{P}(A \times \varnothing)))|=2\).