Given the universal set mu = Z^+ : 1 < x 20 set P = : x is a multiple of 3 and set Q = : x is a positive even number\. Find P Q and its subsets. Also, make x the subject of y = 3k − (2)/(

The universal set $\mu$ consists of counting numbers $x$ such that $1 < x \le 20$.

$$\mu = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}$$

Set $P$ consists of multiples of 3 within $\mu$.

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

Set $Q$ consists of positive even numbers within $\mu$.

$$Q = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}$$

The intersection $P \cap Q$ contains elements common to both $P$ and $Q$.

$$P \cap Q = \{6, 12, 18\}$$

a) (i) The elements in $P \cap Q$ are $\boxed{\{6, 12, 18\}}$.

The set $P \cap Q$ has 3 elements. The number of subsets is $2^3 = 8$.
The subsets are:

$$\emptyset$$ $$\{6\}$$ $$\{12\}$$ $$\{18\}$$ $$\{6, 12\}$$ $$\{6, 18\}$$ $$\{12, 18\}$$ $$\{6, 12, 18\}$$

a) (ii) The subsets in $P \cap Q$ are $\boxed{\{\emptyset, \{6\}, \{12\}, \{18\}, \{6, 12\}, \{6, 18\}, \{12, 18\}, \{6, 12, 18\}\}}$.

$$y = 3k - \frac{2}{x}$$

Subtract $3k$ from both sides:

$$y - 3k = -\frac{2}{x}$$

Multiply both sides by $-1$:

$$3k - y = \frac{2}{x}$$