The set of the counting numbers which are multiple of `6` and less than `50` is written in set builder form as

Options

A. `{x:x in N and "x is a multiple of 6"}`
B. `{x:x=6 and 0 lt x lt 50}`
C. `{x:x " is a multiple of 6" and 0 lt x lt 50, x in N}`
D. `{x:x " is a multiple of 6" and x le 50}`

Answer

To express the set of counting numbers that are multiples of 6 and less than 50 in set builder form, follow these steps:

### Step-by-Step Solution:

1. **Identify the Elements**: We need to find counting numbers that are multiples of 6. Counting numbers start from 1 and go upwards (1, 2, 3, ...).

2. **Define the Condition**: The numbers must be multiples of 6. A number \( x \) is a multiple of 6 if it can be expressed as \( x = 6n \), where \( n \) is a positive integer.

3. **Set the Range**: The problem states that these multiples of 6 should be less than 50. Therefore, we need to ensure that \( x < 50 \).

4. **Combine the Conditions**: We can express the conditions as:
- \( x \) is a multiple of 6 (i.e., \( x = 6n \) for some integer \( n \))
- \( x < 50 \)
- Since we are dealing with counting numbers, we also need \( x > 0 \).

5. **Write in Set Builder Notation**: Combining these conditions, we can write the set in set builder form as:
\[ \{ x \in \mathbb{N} : x = 6n, n \in \mathbb{N}, x < 50 \} \]

### Final Answer:
The set of counting numbers which are multiples of 6 and less than 50 in set builder form is:
\[ \{ x \in \mathbb{N} : x = 6n, n \in \mathbb{N}, x < 50 \} \]