Let P = { x: x is a factor of 18 } and Q = { x : x is a factor of 24}.
Find : (a) ` P uu Q `
(b) ` P nn Q `

Answer

To solve the problem, we need to find the union and intersection of two sets, P and Q, where:

- \( P = \{ x : x \text{ is a factor of } 18 \} \)
- \( Q = \{ x : x \text{ is a factor of } 24 \} \)

### Step 1: Identify the factors of 18
To find the factors of 18, we need to determine which numbers divide 18 without leaving a remainder.

The factors of 18 are:
- 1 (18 ÷ 1 = 18)
- 2 (18 ÷ 2 = 9)
- 3 (18 ÷ 3 = 6)
- 6 (18 ÷ 6 = 3)
- 9 (18 ÷ 9 = 2)
- 18 (18 ÷ 18 = 1)

Thus, we can write:
\[ P = \{ 1, 2, 3, 6, 9, 18 \} \]

### Step 2: Identify the factors of 24
Next, we find the factors of 24 in a similar manner.

The factors of 24 are:
- 1 (24 ÷ 1 = 24)
- 2 (24 ÷ 2 = 12)
- 3 (24 ÷ 3 = 8)
- 4 (24 ÷ 4 = 6)
- 6 (24 ÷ 6 = 4)
- 8 (24 ÷ 8 = 3)
- 12 (24 ÷ 12 = 2)
- 24 (24 ÷ 24 = 1)

Thus, we can write:
\[ Q = \{ 1, 2, 3, 4, 6, 8, 12, 24 \} \]

### Step 3: Find the union of P and Q
The union of two sets, denoted as \( P \cup Q \), includes all elements that are in either set P or set Q (or both).

Combining the elements of both sets without repetition:
\[ P \cup Q = \{ 1, 2, 3, 4, 6, 8, 9, 12, 18, 24 \} \]

### Step 4: Find the intersection of P and Q
The intersection of two sets, denoted as \( P \cap Q \), includes only the elements that are present in both sets P and Q.

Identifying the common elements:
- From set P: \( \{ 1, 2, 3, 6, 9, 18 \} \)
- From set Q: \( \{ 1, 2, 3, 4, 6, 8, 12, 24 \} \)

The common elements are:
\[ P \cap Q = \{ 1, 2, 3, 6 \} \]

### Final Answers
(a) \( P \cup Q = \{ 1, 2, 3, 4, 6, 8, 9, 12, 18, 24 \} \)

(b) \( P \cap Q = \{ 1, 2, 3, 6 \} \)