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If X = 4 n - 3 n - 1 : n ∈ N and Y = 9 n - 1 : n ∈ N , where N is the set of natural numbers, then X ∪ Y is equal to

Options

  1. AX
  2. BY
  3. CN
  4. DY - X

Correct answer

B. Y

Step-by-step solution

X = 4 n - 3 n - 1 : n ∈ N 4 n - 3 n - 1 = 3 + 1 n - 3 n - 1 = n C 0 3 o 1 n + n C 1 3 1 1 n - 1 + . .... + n C n 3 n - 3 n - 1 = 1 + n 3 + n C 2 3 2 + . .... + 3 n - 3 n - 1 = n C 2 3 2 + n C 3 3 3 + . .... + 3 n = 3 2 n C 2 + n C 3 3 1 + n C 4 3 2 + . .... + 3 n - 2 = 9 k , where k is a natural number. Y = 9 n - 1   :  n ∈ N ∴   X ⊆ Y ∴   X ∪ Y = Y

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