Given the universal set mu = ; 1 < x < 20, x is an integer set P = : x is a multiple of 3 and set Q = : x is a prime number find P Q', P' Q, and (P Q)'.

The universal set $\mu = \{x; 1 < x < 20, \text{ where } x \text{ is an integer}\}$.

$$\mu = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}$$

Set $P = \{x: x \text{ is a multiple of } 3\}$.

$$P = \{3, 6, 9, 12, 15, 18\}$$

Set $Q = \{x: x \text{ is a prime number}\}$.

$$Q = \{2, 3, 5, 7, 11, 13, 17, 19\}$$

$P'$ contains all elements in $\mu$ that are not in $P$.

$$P' = \{2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19\}$$

$Q'$ contains all elements in $\mu$ that are not in $Q$.

$$Q' = \{4, 6, 8, 9, 10, 12, 14, 15, 16, 18\}$$

Now we can solve parts a), b), and c).

a) Find $P \cap Q'$
This is the intersection of set $P$ and set $Q'$, meaning elements that are in both $P$ and $Q'$.

$$P \cap Q' = \{3, 6, 9, 12, 15, 18\} \cap \{4, 6, 8, 9, 10, 12, 14, 15, 16, 18\}$$ $$P \cap Q' = \boxed{\{6, 9, 12, 15, 18\}}$$

b) Find $P' \cup Q$
This is the union of set $P'$ and set $Q$, meaning all elements that are in $P'$ or in $Q$ (or both).

$$P' \cup Q = \{2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19\} \cup \{2, 3, 5, 7, 11, 13, 17, 19\}$$ $$P' \cup Q = \boxed{\{2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19\}}$$

c) Find $(P \cup Q)'$
First, find the union of $P$ and $Q$.

$$P \cup Q = \{3, 6, 9, 12, 15, 18\} \cup \{2, 3, 5, 7, 11, 13, 17, 19\}$$ $$P \cup Q = \{2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19\}$$

Next, find the complement of $(P \cup Q)$, which means elements in $\mu$ that are not in $P \cup Q$.

$$(P \cup Q)' = \mu \setminus (P \cup Q)$$ $$(P \cup Q)' = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\} \setminus \{2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19\}$$ $$(P \cup Q)' = \boxed{\{4, 8, 10, 14, 16\}}$$