Which of the following are empty sets?Justify.
6. `A={x:x in N and 3 lt x lt 4}`

Answer

To determine whether the set \( A = \{ x : x \in \mathbb{N} \text{ and } 3 < x < 4 \} \) is an empty set, we will analyze the conditions defined for the elements of the set.

### Step 1: Understand the definition of the set
The set \( A \) is defined as the collection of all \( x \) such that \( x \) is a natural number and \( x \) is greater than 3 but less than 4.

### Step 2: Identify the natural numbers
The natural numbers \( \mathbb{N} \) are defined as the set of positive integers: \( \{ 1, 2, 3, 4, 5, \ldots \} \).

### Step 3: Analyze the range given in the set
We need to find natural numbers \( x \) that satisfy the condition \( 3 < x < 4 \).

### Step 4: Check for natural numbers in the range
- The only natural number greater than 3 is 4.
- However, 4 is not less than 4. Therefore, there are no natural numbers that satisfy the condition \( 3 < x < 4 \).

### Step 5: Conclusion
Since there are no natural numbers that meet the criteria specified in the set \( A \), we conclude that \( A \) is an empty set.

Thus, we can write:
\[ A = \emptyset \]

### Final Answer
The set \( A \) is an empty set because there are no natural numbers that satisfy the condition \( 3 < x < 4 \).

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