6 is an even natural number and it is divisible by 3. Therefore 6 belong to the set x. Hence, X is not an empty set. 

It is not an empty set....beacuse there r even natural numbers divisible by 3 like 6 ,24, etc

Explanation:

To prove that the set X containing even natural numbers divisible by 3 is an empty set, we need to demonstrate that there are no elements that satisfy these conditions.

Understanding the Problem:

We are given that X contains even natural numbers divisible by 3. Let's break down the information provided:

- Even natural numbers: These are positive integers that can be divided evenly by 2.
- Divisible by 3: These numbers are evenly divisible by 3 without any remainder.

Analysis:

To find the elements in set X, we need to identify the even natural numbers that are divisible by 3. However, no such numbers exist. Let's analyze why:

1. Even natural numbers: Every even natural number can be represented as 2n, where n is a positive integer. For example, 2, 4, 6, 8, etc.

2. Divisible by 3: If a number is divisible by 3, the sum of its digits must also be divisible by 3.

Proof:

Let's assume that there is an even natural number x that is divisible by 3. We can represent it as 2n, where n is a positive integer.

The sum of the digits of x would be equal to the sum of the digits of 2n. However, the sum of the digits of 2n would be equal to the sum of the digits of n, multiplied by 2.

If n is a positive integer, the sum of its digits will always be greater than or equal to n itself. Therefore, the sum of the digits of 2n will be greater than or equal to 2n.

Since the sum of the digits of 2n is not divisible by 3, it contradicts our assumption that x is divisible by 3.

Therefore, there are no even natural numbers divisible by 3, and the set X is empty.

Conclusion:

In conclusion, the set X containing even natural numbers divisible by 3 is an empty set because there are no numbers that satisfy both conditions. The proof demonstrates that no even natural number can be divisible by 3, leading to the empty set X.