For each of the following pairs of sets , identify the disjoint and overlapping sets:
C= {x: x ` in N, x lt 10 } and D = {x: x in N , ` x is a multiple of 5}

Answer

To solve the problem of identifying the disjoint and overlapping sets for the given sets C and D, we will follow these steps:

### Step 1: Define Set C
Set C is defined as:
\[ C = \{ x : x \in \mathbb{N}, x < 10 \} \]
This means we need to list all natural numbers less than 10. The natural numbers less than 10 are:
\[ C = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \]

### Step 2: Define Set D
Set D is defined as:
\[ D = \{ x : x \in \mathbb{N}, x \text{ is a multiple of } 5 \} \]
The multiples of 5 in the natural numbers are:
\[ D = \{ 0, 5, 10, 15, \ldots \} \]
However, since we are only interested in the natural numbers, we can consider:
\[ D = \{ 5, 10, 15, \ldots \} \]
But for our comparison, we will only consider the multiples of 5 that are less than 10:
\[ D = \{ 5 \} \]

### Step 3: Identify Overlapping Elements
Now, we will find the intersection of sets C and D to identify any overlapping elements:
\[ C \cap D = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \cap \{ 5 \} \]
The common element is:
\[ C \cap D = \{ 5 \} \]

### Step 4: Determine if the Sets are Disjoint or Overlapping
- **Disjoint Sets**: Two sets are disjoint if they have no elements in common. In this case, since \( C \cap D \) contains the element 5, the sets C and D are not disjoint.
- **Overlapping Sets**: Since there is at least one common element (5), the sets C and D are overlapping.

### Final Conclusion
- **Set C**: \( \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \)
- **Set D**: \( \{ 5 \} \)
- **C and D are overlapping sets** since they share the element 5.