Maths · Union, intersection and complement of sets and their algebraic properties
If and : \boldsymbol{n} \in \boldsymbol{N}\}, \) where is the set of natural num
If \( \boldsymbol{X}=\left\{4^{n}-3 n-1: n \in N\right\} \) and \( \boldsymbol{Y}=\{\mathbf{9}(\boldsymbol{n}-\mathbf{1}): \boldsymbol{n} \in \boldsymbol{N}\}, \) where \( \mathrm{N} \) is the set of natural numbers, then \( \boldsymbol{X} \cup \boldsymbol{Y} \) is equal to:
- A. \( Y \)
- B. \( N \)
- C. \( Y-x \)
- D. \( x \)
Step-by-step solution
X = {4^n - 3n - 1 : n ∈ N}. For n=1,2,3,... we get 0,9,54,243,... Observe that 4^n - 3n - 1 is divisible by 9 for all n ∈ N (by modulo 9: 4^n ≡ 4,7,1 cyclically, and -3n-1 ≡ -3n-1, sum ≡ 0 mod 9). Thus X ⊆ {multiples of 9}. Y = {9(n-1) : n ∈ N} = {0,9,18,27,...} is the set of all multiples of 9. Hence X ⊆ Y, so X ∪ Y = Y.
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