Question

Write the following sets by listing their elements between braces. $$ \mathscr{P}(\\{a, b\\}) \times \mathscr{P}(\\{0,1\\}) $$

Short answer

The Cartesian product of the power sets of the given sets is \{ (\{}, \{}\), (\{}, \{0\}), (\{}, \{1\}), (\{}, \{0, 1\}), (\{a\}, \{}\), (\{a\}, \{0\}), (\{a\}, \{1\}), (\{a\}, \{0, 1\}), (\{b\}, \{}\), (\{b\}, \{0\}), (\{b\}, \{1\}), (\{b\}, \{0, 1\}), (\{a, b\}, \{}\), (\{a, b\}, \{0\}), (\{a, b\}, \{1\}), (\{a, b\}, \{0, 1\}) \}

Step-by-step solution

Identify the power set of \( \{a, b\} \)

The power set \( \mathscr{P}(\{a, b\}) \) is the set of all subsets of \{a, b\} including the empty set. It is \{\{}, \{a\}, \{b\}, \{a, b\}\}.

Identify the power set of \( \{0,1\} \)

The power set \( \mathscr{P}(\{0,1\}) \) is the set of all subsets of \{0, 1\} including the empty set. This set is \{\{}, \{0\}, \{1\}, \{0, 1\}\}

Take the Cartesian product of the power sets

We combine each element of \( \mathscr{P}(\{a, b\}) \) with each element of \( \mathscr{P}(\{0,1\}) \) to form the Cartesian product, denoted as \( \mathscr{P}(\{a, b\}) \times \mathscr{P}(\{0,1\}) \). This gives us: \{ (\{}, \{}\), (\{}, \{0\}), (\{}, \{1\}), (\{}, \{0, 1\}), (\{a\}, \{}\), (\{a\}, \{0\}), (\{a\}, \{1\}), (\{a\}, \{0, 1\}), (\{b\}, \{}\), (\{b\}, \{0\}), (\{b\}, \{1\}), (\{b\}, \{0, 1\}), (\{a, b\}, \{}\), (\{a, b\}, \{0\}), (\{a, b\}, \{1\}), (\{a, b\}, \{0, 1\}) \}