Let U = x | 18 <= x <= 36. Let P be the set of multiples of 3 in U. Let Q be the set of factors of 72 in U. Let R be the set of even numbers in U. Find P intersect Q and P' intersect R.

The universal set $U = \{x \mid 18 \le x \le 36\}$ includes all integers from 18 to 36, inclusive.

$$U = \{18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36\}$$

Set P consists of multiples of 3 within U.

$$P = \{18, 21, 24, 27, 30, 33, 36\}$$

a.i) $\boxed{\text{}P = \{18, 21, 24, 27, 30, 33, 36\}}$

Set Q consists of factors of 72 within U. The factors of 72 are $\{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\}$.

$$Q = \{18, 24, 36\}$$

a.ii) $\boxed{\text{}Q = \{18, 24, 36\}}$

Set R consists of even numbers within U.

$$R = \{18, 20, 22, 24, 26, 28, 30, 32, 34, 36\}$$

a.iii) $\boxed{\text{}R = \{18, 20, 22, 24, 26, 28, 30, 32, 34, 36\}}$

P $\cap$ Q represents the intersection of sets P and Q, containing elements common to both sets.

$$P \cap Q = \{18, 24, 36\}$$

a.iv) $\boxed{\text{}P \cap Q = \{18, 24, 36\}}$

First, find P', the complement of P. P' contains all elements in U that are not in P.

$$P' = U - P = \{19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35\}$$

Next, find P' $\cap$ R, the intersection of P' and R.

$$P' \cap R = \{20, 22, 26, 28, 32, 34\}$$

a.v) $\boxed{\text{}P' \cap R = \{20, 22, 26, 28, 32, 34\}}$