1. **Problem statement:** Given the universal set $\varepsilon = \{x : x \text{ is a positive integer}\}$, and sets $P = \{x : x < 9\}$ and $Q = \{x : x > 4\}$, we need to: a. List the elements of $P \cap Q$. b. Find $n(Q')$, the number of elements in the complement of $Q$. 2. **Recall definitions and formulas:** - The intersection $P \cap Q$ contains elements common to both $P$ and $Q$. - The complement $Q'$ contains elements in $\varepsilon$ that are not in $Q$. - Since $\varepsilon$ is the set of all positive integers, it is infinite. 3. **Find $P \cap Q$:** - $P = \{x : x < 9\} = \{1, 2, 3, 4, 5, 6, 7, 8\}$. - $Q = \{x : x > 4\} = \{5, 6, 7, 8, 9, 10, 11, \ldots\}$. - The intersection $P \cap Q$ is the set of $x$ such that $x < 9$ and $x > 4$. - So, $P \cap Q = \{5, 6, 7, 8\}$. 4. **Find $n(Q')$:** - $Q' = \{x \in \varepsilon : x \leq 4\}$ because $Q$ contains all positive integers greater than 4. - So, $Q' = \{1, 2, 3, 4\}$. - The number of elements in $Q'$ is $n(Q') = 4$. **Final answers:** - $P \cap Q = \{5, 6, 7, 8\}$ - $n(Q') = 4$