State whether the following sets are empty finite or infinite sets., mention the cardinal number.
{x:x `in` N and x is a composite number `lt12` }

Answer

To determine whether the set \( \{ x : x \in \mathbb{N} \text{ and } x \text{ is a composite number} < 12 \} \) is empty, finite, or infinite, and to find its cardinal number, we can follow these steps:

### Step 1: Understand the Definitions
- **Natural Numbers (\(\mathbb{N}\))**: These are the counting numbers starting from 1, i.e., \(1, 2, 3, 4, \ldots\).
- **Composite Numbers**: These are numbers that have more than two distinct positive divisors (factors). For example, the number 4 is composite because it can be divided by 1, 2, and 4.

### Step 2: Identify Composite Numbers Less Than 12
We need to find the composite numbers that are less than 12. We can list the natural numbers less than 12:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \]

Now, we will identify which of these numbers are composite:
- **4**: Factors are 1, 2, 4 (composite)
- **6**: Factors are 1, 2, 3, 6 (composite)
- **8**: Factors are 1, 2, 4, 8 (composite)
- **9**: Factors are 1, 3, 9 (composite)
- **10**: Factors are 1, 2, 5, 10 (composite)

The composite numbers less than 12 are:
\[ 4, 6, 8, 9, 10 \]

### Step 3: Form the Set
Now we can form the set of composite numbers less than 12:
\[ \{ 4, 6, 8, 9, 10 \} \]

### Step 4: Determine if the Set is Empty, Finite, or Infinite
Since we have identified specific numbers in the set, it is not empty. Furthermore, the set contains a finite number of elements (5 elements).

### Step 5: Find the Cardinal Number
The cardinal number of a set is the number of elements in that set. Here, we have:
\[ \text{Cardinal Number} = 5 \]

### Final Answer
The set \( \{ x : x \in \mathbb{N} \text{ and } x \text{ is a composite number} < 12 \} \) is a **finite set** with a **cardinal number of 5**.

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