Question

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

Short answer

The power set \( \mathscr{P}(\{a, b\} \times \{0\}) \) is \( \{\emptyset , \{(a,0)\}, \{(b,0)\}, \{(a,0), (b,0)\} \} \)

Step-by-step solution

Understand the Cartesian Product

The Cartesian product, denoted with \( \times \), of two sets is a set in which each element is an ordered pair that comes from the two sets. So, \( \{a, b\} \times \{0\} = \{(a,0), (b,0)\} \). Here, each pair is (first element from set A, second element from set B).

Understand the Power Set

The power set of a set includes all possible subsets of that set. It also includes the empty set and the set itself.

Compute the Power Set

To compute the power set of \( \{(a,0), (b,0)\} \), one has to list all subsets. It should include the empty set, the set itself, all the sets with a single element, and all the sets with two elements. The power set, therefore, is \( \mathscr{P}(\{(a,0), (b,0)\}) = \{\emptyset , \{(a,0)\}, \{(b,0)\}, \{(a,0), (b,0)\} \} \).