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

Engineering

Mathematics

Sets

Question

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

Y – X

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X = {0, 9, 54, 243 ……, ……}

Y = {0, 9, 18, 27……, ……}

X $\cup$ Y = Y.

[Note that : 4ⁿ – 3n – 1 = (1 + 3)ⁿ – 3n – 1

= (1 + 3n + ⁿC₂ · 3² + ⁿC₃ · 3³ + ……)

– 3n – 1 = ⁿC₂ · 3² + ⁿC₃ · 3³ + ……,

which is multiple of 9.

Here, X $\cap$ Y = X.