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

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 ‘$X \cup Y$’.