Given the universal set U = 1, ..., 18, find the following sets: a) A ∪ B', b) A ∩ C ∩ D, c) B ∩ (C ∪ D)' where A is even, B is less than 10, C is a multiple of 3, and D is prime.

Here are the elements of each set based on the given definitions:
The universal set is $\mathcal{U} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18\}$.

• $A = \{x : x \in \mathcal{U} \text{ and } x \text{ is even}\}$ $A = \{2, 4, 6, 8, 10, 12, 14, 16, 18\}$

• $B = \{x : x \in \mathcal{U} \text{ and } x \text{ is less than } 10\}$ $B = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$

• $C = \{x : x \in \mathcal{U} \text{ and } x \text{ is a multiple of } 3\}$ $C = \{3, 6, 9, 12, 15, 18\}$

• $D = \{x : x \in \mathcal{U} \text{ and } x \text{ is a prime number}\}$ $D = \{2, 3, 5, 7, 11, 13, 17\}$

Now, let's solve each part:

a) $A \cup B'$

Step 1: Find $B'$.
$B' = \mathcal{U} - B = \{10, 11, 12, 13, 14, 15, 16, 17, 18\}$

Step 2: Find $A \cup B'$.
$A \cup B' = \{2, 4, 6, 8, 10, 12, 14, 16, 18\} \cup \{10, 11, 12, 13, 14, 15, 16, 17, 18\}$

$$A \cup B' = \boxed{\{2, 4, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18\}}$$

b) $A \cap C \cap D$

Step 1: Find $A \cap C$.
$A \cap C = \{2, 4, 6, 8, 10, 12, 14, 16, 18\} \cap \{3, 6, 9, 12, 15, 18\}$
$A \cap C = \{6, 12, 18\}$

Step 2: Find $(A \cap C) \cap D$.
$(A \cap C) \cap D = \{6, 12, 18\} \cap \{2, 3, 5, 7, 11, 13, 17\}$

$$(A \cap C) \cap D = \boxed{\emptyset}$$

c) $B \cap (C \cup D)'$

Step 1: Find $C \cup D$.
$C \cup D = \{3, 6, 9, 12, 15, 18\} \cup \{2, 3, 5, 7, 11, 13, 17\}$
$C \cup D = \{2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18\}$

Step 2: Find $(C \cup D)'$.
$(C \cup D)' = \mathcal{U} - (C \cup D) = \{1, 4, 8, 10, 14, 16\}$

Step 3: Find $B \cap (C \cup D)'$.
$B \cap (C \cup D)' = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \cap \{1, 4, 8, 10, 14, 16\}$

$$B \cap (C \cup D)' = \boxed{\{1, 4, 8\}}$$

d) $(A \cup B') - B$

Step 1: From part a), $A \cup B' = \{2, 4, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18\}$.

Step 2: Find $(A \cup B') - B$.
$(A \cup B') - B = \{2, 4, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18\} - \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$

$$(A \cup B') - B = \boxed{\{10, 11, 12, 13, 14, 15, 16, 17, 18\}}$$

e) $(A - B)'$

Step 1: Find $A - B$.
$A - B = \{2, 4, 6, 8, 10, 12, 14, 16, 18\} - \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$
$A - B = \{10, 12, 14, 16, 18\}$

Step 2: Find $(A - B)'$.
$(A - B)' = \mathcal{U} - (A - B) = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17\}$

$$(A - B)' = \boxed{\{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17\}}$$

f) $|C \cup D|$

Step 1: From part c), $C \cup D = \{2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18\}$.

Step 2: Count the number of elements in $C \cup D$.

$$|C \cup D| = \boxed{12}$$

g) $(A - B)' \cap A$

Step 1: From part e), $(A - B)' = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17\}$.

Step 2: Find $(A - B)' \cap A$.
$(A - B)' \cap A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17\} \cap \{2, 4, 6, 8, 10, 12, 14, 16, 18\}$

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