Question:
If $X = \{4^n- 3 n - 1: n \in N \}$ and $Y = \{9 (n - 1): n \in N \}$ ,where $N$ is the set of natural numbers, then $ X \cup Y$ is equal to
If $X = \{4^n- 3 n - 1: n \in N \}$ and $Y = \{9 (n - 1): n \in N \}$ ,where $N$ is the set of natural numbers, then $ X \cup Y$ is equal to
Updated On: Feb 14, 2025
- N
- Y-X
- X
- Y
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The Correct Option is D
Solution and Explanation
$X=\left\{(1+3)^{n}-3 n-1, n \in N\right\}$
$=3^{2}({ }^{n} C_{2}+{ }^{n} C_{3} \cdot 3+\ldots+3^{n-2}), n \in N\} $
$=\{\text { Divisible by } 9\}$
$Y=\{9(n-1), n \in N\} $
$=(\text { All multiples of } 9\} $
So, $X \subseteq Y $
i.e., $X \cup Y=Y $
$=3^{2}({ }^{n} C_{2}+{ }^{n} C_{3} \cdot 3+\ldots+3^{n-2}), n \in N\} $
$=\{\text { Divisible by } 9\}$
$Y=\{9(n-1), n \in N\} $
$=(\text { All multiples of } 9\} $
So, $X \subseteq Y $
i.e., $X \cup Y=Y $
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".