For each of the following pairs of sets , identify the disjoint and overlapping sets:
E = {x: x 4n, ` n in N ` } and F = {x: x = 9 n , ` n in N`}

Answer

To determine whether the sets E and F are disjoint or overlapping, we will follow these steps:

### Step 1: Define the Sets
- Set E is defined as \( E = \{ x : x = 4n, n \in \mathbb{N} \} \)
- Set F is defined as \( F = \{ x : x = 9n, n \in \mathbb{N} \} \)

### Step 2: Generate Elements of Set E
To find the elements of set E, we will substitute natural numbers (1, 2, 3, ...) into the expression \( 4n \):
- For \( n = 1 \): \( x = 4 \times 1 = 4 \)
- For \( n = 2 \): \( x = 4 \times 2 = 8 \)
- For \( n = 3 \): \( x = 4 \times 3 = 12 \)
- For \( n = 4 \): \( x = 4 \times 4 = 16 \)
- For \( n = 5 \): \( x = 4 \times 5 = 20 \)
- Continuing this way, we get: \( E = \{ 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ... \} \)

### Step 3: Generate Elements of Set F
Now, we will generate elements of set F by substituting natural numbers into the expression \( 9n \):
- For \( n = 1 \): \( x = 9 \times 1 = 9 \)
- For \( n = 2 \): \( x = 9 \times 2 = 18 \)
- For \( n = 3 \): \( x = 9 \times 3 = 27 \)
- For \( n = 4 \): \( x = 9 \times 4 = 36 \)
- For \( n = 5 \): \( x = 9 \times 5 = 45 \)
- Continuing this way, we get: \( F = \{ 9, 18, 27, 36, 45, 54, ... \} \)

### Step 4: Identify Common Elements
Now, we will check for common elements between sets E and F:
- From set E, we have elements like \( 4, 8, 12, 16, 20, 24, 28, 32, 36, ... \)
- From set F, we have elements like \( 9, 18, 27, 36, 45, ... \)

The common element found in both sets is \( 36 \).

### Step 5: Conclusion
Since there is at least one common element (36) between sets E and F, we conclude that:
- Sets E and F are **overlapping sets**.

### Summary
- Set E: \( \{ 4, 8, 12, 16, 20, 24, ... \} \)
- Set F: \( \{ 9, 18, 27, 36, 45, ... \} \)
- E and F are overlapping sets.

---