Choose the correct way to write the set in set-builder notation. $$ A=\{3,6,9,12, \ldots\} $$ $$ \mathbf{A}=\{ $$ numbers that are multiples of 3$$ \} $$ $$ \mathbf{A}=\{x \mid x $$ is an odd integer $$ \} $$ $$ \mathbf{A}=\{x $$ is a multiple of a natural number and 3$$ \} $$ $$ \mathbf{A}=\{x \mid x \in N $$ and x is a multiple of 3$$ \} $$ Check Answer

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The correct way to write the set $$ A = \{3, 6, 9, 12, \ldots\} $$ in set-builder notation is:

$$ \mathbf{A} = \{\, x \mid x \in \mathbb{N} \text{ and } x \text{ is a multiple of } 3 \,\} $$

**Explanation:**

- **$$ \mathbf{A} = \{ x \mid x \in \mathbb{N} \text{ and } x \text{ is a multiple of } 3 \} $$**  
  This notation correctly specifies that $$ x $$ belongs to the set of natural numbers ($$ \mathbb{N} $$) and that $$ x $$ is a multiple of 3. This precisely captures the elements of $$ A $$.

**Why the Other Options Are Incorrect:**

1. **$$ \mathbf{A} = \{ \text{numbers that are multiples of 3} \} $$**  
   This is vague because it doesn't specify that the numbers are natural numbers. It could include negative multiples or zero, which are not in the original set $$ A $$.

2. **$$ \mathbf{A} = \{ x \mid x \text{ is an odd integer} \} $$**  
   This is incorrect because the set $$ A $$ includes both odd and even multiples of 3 (e.g., 6, 12 are even).

3. **$$ \mathbf{A} = \{ x \mid x \text{ is a multiple of a natural number and 3} \} $$**  
   This wording is unclear and doesn't precisely define the relationship between $$ x $$, natural numbers, and 3.

Therefore, the most accurate and precise set-builder notation for $$ A $$ is:

$$ \mathbf{A} = \{\, x \mid x \in \mathbb{N} \text{ and } x \text{ is a multiple of } 3 \,\} $$
The correct set-builder notation for $$ A = \{3, 6, 9, 12, \ldots\} $$ is: $$ \mathbf{A} = \{\, x \mid x \in \mathbb{N} \text{ and } x \text{ is a multiple of } 3 \,\} $$

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