Question

If \[X = \left\{ {{4^n} - 3n - 1:n\,belons\,to\,N} \right\}\] and \[Y = \left\{ {9\left( {n - 1} \right):n\,belons\,to\,N} \right\}\], 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

Hint:Here in this question based on set theory, given the two sets \[X\] and \[Y\] belong to the set of natural numbers \[N\]. Clearly, we can tell the given set \[Y\] is a multiple of 9 and values of \[X\] set can be found by giving \[n\] values as \[n = \]0, 1, 2, 3, …. Depending upon the nature of set \[X\] and the concept of sets we can find the union of set \[X\] and \[Y\] i.e., \[X \cup Y\].

Complete step by step answer:
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

Consider the question: Given, if \[X = \left\{ {{4^n} - 3n - 1:n\,belons\,to\,N} \right\}\] and \[Y = \left\{ {9\left( {n - 1} \right):n\,belons\,to\,N} \right\}\] where \[n\] belongs to the set of all natural number \[N\] i.e., \[n = \]0, 1, 2, 3, ….
Let us take \[X = {4^n} - 3n - 1\].At,
\[n = 1\], \[X = {4^1} - 3\left( 1 \right) - 1 = 4 - 3 - 1 = 0\]
\[\Rightarrow n = 2\], \[X = {4^2} - 3\left( 2 \right) - 1 = 16 - 6 - 1 = 9\]
\[\Rightarrow n = 3\], \[X = {4^3} - 3\left( 3 \right) - 1 = 64 - 9 - 1 = 54\]
\[\Rightarrow n = 4\], \[X = {4^4} - 3\left( 4 \right) - 1 = 256 - 12 - 1 = 243\] and so on
Therefore, \[X\] can take Values \[X = \left\{ {0,9,54,243,....} \right\}\] when \[n = 1,2,3,....\].

Let us take \[Y = 9\left( {n - 1} \right)\]. At
\[n = 1\], \[Y = 9\left( {1 - 1} \right) = 9\left( 0 \right) = 0\]
\[\Rightarrow n = 2\], \[Y = 9\left( {2 - 1} \right) = 9\left( 1 \right) = 9\]
\[\Rightarrow n = 3\], \[Y = 9\left( {3 - 1} \right) = 9\left( 2 \right) = 18\]
\[\Rightarrow n = 4\], \[Y = 9\left( {4 - 1} \right) = 9\left( 3 \right) = 27\] and so on
Therefore, \[Y\] can take Values \[Y = \left\{ {0,9,18,27,....} \right\}\] when \[n = 1,2,3,....\]
The set \[Y\] represents multiples of 9. On observing the set \[X\] and \[Y\] we can say clearly, \[X\] is a subset of \[Y\] i.e., \[X \subset Y\]. It means the set \[Y\] is a bigger set; it includes the elements of set \[X\]. Hence, \[X \cup Y = Y\]

Therefore, option D is the correct answer.

Note:To solve the problem based on set theory. Students must know the components of set theory like subset, union, intersection etc. Remember subset is a set of which all the elements are contained in another set and it’s denoted by ‘\[ \subset \]’ and the union of two sets X and Y is equal to the set of elements which are present in both the sets X and Y and it’s denoted by ‘\[X \cup Y\]’.