Question

Write the following sets by listing their elements between braces. $$ \\{X \in \mathscr{P}(\\{1,2,3\\}): 2 \in X\\} $$

Short answer

The given set \{X \in \mathscr{P}(\{1,2,3\}): 2 \in X\} can be written by listing its elements as \{\{2\}, \{1,2\}, \{2,3\}, \{1,2,3\}\}.

Step-by-step solution

Identify all possible subsets

First, let's find all the possible subsets (also known as power set) of the set \({1,2,3}\). The power set of a set S, denoted by \(\mathscr{P}(S)\), is the set of all possible subsets of S. So for S = \({1,2,3}\), the subsets and thus the power set would be \(\mathscr{P}(S) = \{\{\}, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}\)

Identify the subsets that include 2

Next, we need to find the subsets from the power set \(\mathscr{P}(S)\) that include 2. It is clear that the subsets with 2 as a member are \{2\}, \{1,2\}, \{2,3\}, \{1,2,3\}.