For each of the following pairs of sets , identify the disjoint and overlapping sets:
G= { x: x = 8n, ` n in N and n lt 7 } and H= {x: x = 9 n , n in N and n lt 7 }`

Answer

To solve the problem of identifying whether the sets G and H are disjoint or overlapping, we will follow these steps:

### Step 1: Define the Sets
We need to define the sets G and H based on the given conditions.

- **Set G**: G = { x : x = 8n, n ∈ N and n < 7 }
- **Set H**: H = { x : x = 9n, n ∈ N and n < 7 }

### Step 2: Determine the Elements of Set G
We will find the elements of set G by substituting values of n from 1 to 6 (since n is a natural number less than 7):

- For n = 1: x = 8 * 1 = 8
- For n = 2: x = 8 * 2 = 16
- For n = 3: x = 8 * 3 = 24
- For n = 4: x = 8 * 4 = 32
- For n = 5: x = 8 * 5 = 40
- For n = 6: x = 8 * 6 = 48

Thus, the elements of set G are:
\[ G = \{ 8, 16, 24, 32, 40, 48 \} \]

### Step 3: Determine the Elements of Set H
Next, we will find the elements of set H by substituting values of n from 1 to 6:

- For n = 1: x = 9 * 1 = 9
- For n = 2: x = 9 * 2 = 18
- For n = 3: x = 9 * 3 = 27
- For n = 4: x = 9 * 4 = 36
- For n = 5: x = 9 * 5 = 45
- For n = 6: x = 9 * 6 = 54

Thus, the elements of set H are:
\[ H = \{ 9, 18, 27, 36, 45, 54 \} \]

### Step 4: Compare the Sets
Now we will compare the elements of sets G and H to determine if they are disjoint or overlapping.

- Set G: { 8, 16, 24, 32, 40, 48 }
- Set H: { 9, 18, 27, 36, 45, 54 }

### Step 5: Identify Overlapping or Disjoint Sets
We check if there are any common elements between sets G and H.

- The elements of G are: 8, 16, 24, 32, 40, 48
- The elements of H are: 9, 18, 27, 36, 45, 54

Since there are no common elements between the two sets, we conclude that:

**G and H are disjoint sets.**

### Final Answer
G and H are disjoint sets.

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