2. Write the following sets by listing their elements between braces: (a) \( \mathcal{P}(\{1,2\} \times\{3\}) \) (b) \( \mathcal{P}(\{1,2,3,4\}) \) (c) If \( |A|=m \) and \( |B|=n \), what is \( |\mathcal{P}(A \times B)| \) ?

The first set, \( \mathcal{P}(\{1,2\} \times\{3\}) \), comprises the subsets of the Cartesian product \( \{1,2\} \times\{3\} \), which results in \( \{(1,3), (2,3)\} \). Its power set includes: \( \{\}, \{(1,3)\}, \{(2,3)\}, \{(1,3), (2,3)\} \), so we can express this as \( \mathcal{P}(\{(1,3), (2,3)\}) = \{\emptyset, \{(1,3)\}, \{(2,3)\}, \{(1,3), (2,3)\}\} \). For the second set, \( \mathcal{P}(\{1,2,3,4\}) \), we'll note that the power set contains all subsets of the set \( \{1,2,3,4\} \). Therefore, it will include: \( \{\}, \{1\}, \{2\}, \{3\}, \{4\}, \{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}, \{3,4\}, \{1,2,3\}, \{1,2,4\}, \{1,3,4\}, \{2,3,4\}, \{1,2,3,4\} \). This can be listed as \( \mathcal{P}(\{1,2,3,4\}) = \{ \emptyset, \{1\}, \{2\}, \{3\}, \{4\}, \{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}, \{3,4\}, \{1,2,3\}, \{1,2,4\}, \{1,3,4\}, \{2,3,4\}, \{1,2,3,4\} \} \). Lastly, using basic principles of set theory, if \( |A|=m \) and \( |B|=n \), the Cartesian product \( A \times B \) will have \( m \times n \) elements. The power set \( \mathcal{P}(A \times B) \) will thus contain \( 2^{m \times n} \) subsets, meaning \( |\mathcal{P}(A \times B)| = 2^{m \cdot n} \).