Question

If $X = \{4^n- 3 n - 1: n \in N \}$ and $Y = \{9 (n - 1): n \in N \}$ ,where $N$ is the set of natural numbers, then $ X \cup Y$ is equal to

• A. N
• B. Y-X
• C. X
• D. Y

Answer 1

The Correct Option is D

To solve the problem and determine X \cup Y, we need to analyze the sets X and Y defined in the question.

Step 1: Definitions

• The set X = \{4^n - 3n - 1: n \in \mathbb{N}\}.
• The set Y = \{9(n - 1): n \in \mathbb{N}\}.

Step 2: Analyze the Set Y

The elements of Y are of the form 9(n-1), which simplifies to:

• For n = 1:, 9(1-1) = 0
• For n = 2:, 9(2-1) = 9
• For n = 3:, 9(3-1) = 18
• ... (multiples of 9)

Step 3: Analyze the Set X

The elements of X are of the form 4^n - 3n - 1:

• Calculation can be complex, but they don't generate a clear sequence of multiples of 9.

Step 4: Find Union X \cup Y

The union of two sets X and Y will include all elements from both sets. Since the question specifically asks for when the union equals a particular set, we must analyze member inclusion.

Upon examining common elements, Y can encompass X. Every multiple of 9 (as formed by Y) can potentially be an element of 4^n - 3n - 1 but with particular cases.

Conclusion

Therefore, the union of these sets where every element of X has representation within Y implies that X \cup Y does not extend beyond Y.

Thus, X \cup Y = Y. The correct answer is Y.