Question

Let \[X = x\left\{ {x:x = {n^3} + 2n + 1,n \in R} \right\}\]and \[Y = \left\{ {x:x = 3{n^2} + 7,n \in R} \right\}\]then \[X \cap Y\]is a subset of
A.\[\left\{ {x:x = 3n + 5,n \in R} \right\}\]
B.\[\left\{ {x:x = {n^2} + n + 1,n \in R} \right\}\]
C.\[\left\{ {x:x = 7n - 1,n \in R} \right\}\]
D.None of the above

Answer 1

Hint: A set is a collection of objects where each object in the set is called an element for that set denoted by \[x \in R\]where x is the element of the set R and a set having no elements to them is called an empty set.

Complete step-by-step answer:
An interval is a set number that consists of all real numbers between a given pair of numbers. An endpoint of an interval is either of the two points that mark the endpoint of a line segment. An interval can be of different types which can include either endpoint or both endpoints or neither endpoint.
An interval that does not include endpoints is an open interval denoted by round brackets (). For closed intervals, they include endpoints of the interval, and they are denoted by square brackets []. Any interval which includes either of the endpoints is denoted by (].
In this question X and Y are two sets whose sub-set is n, which includes real numbers, we need to find the value of both the sets by substituting the value of n in \[X \cap Y\] subset and check for the options.
\[X = x\left\{ {x:x = {n^3} + 2n + 1,n \in R} \right\}\]
\[Y = \left\{ {x:x = 3{n^2} + 7,n \in R} \right\}\]
The data of the set X where the number n is a natural number\[\left\{ {n = 1,2,3,4,5,6.7....} \right\}\]
 \[
  X = x\left\{ {x:x = {n^3} + 2n + 1,n \in R} \right\} \\
   = \left\{ {4,13,34,73,136.....} \right\} \\
 \]
\[
  Y = \left\{ {x:x = 3{n^2} + 7,n \in R} \right\} \\
   = \left\{ {10,19,34,55,82,115....} \right\} \\
 \]
Hence, one of the values of the subset is: \[X \cap Y = \left\{ {34,...} \right\}\]
Now proceed by checking the options which should contain the value of 34 in its range as:
\[\left\{ {x:x = 3n + 5,n \in R} \right\}\]
\[\left\{ {x:x = 3n + 5,n \in R} \right\} = \left\{ {8,11,14,17,20,23....} \right\}\]
Does not include 34, hence, \[\left\{ {x:x = 3n + 5,n \in R} \right\}\] is not a subset of \[X \cap Y\].
\[\left\{ {x:x = {n^2} + n + 1,n \in R} \right\}\]
\[\left\{ {x:x = {n^2} + n + 1,n \in R} \right\} = \left\{ {4,7,13,21,31,....} \right\}\]
Does not include 34, hence, \[\left\{ {x:x = 3n + 5,n \in R} \right\}\] is not a subset of \[X \cap Y\].
\[\left\{ {x:x = 7n - 1,n \in R} \right\}\]
\[\left\{ {x:x = 7n - 1,n \in R} \right\} = \left\{ {6,13,20,27,34,41.....} \right\}\]
Includes 34, hence, \[\left\{ {x:x = 3n + 5,n \in R} \right\}\] is not a subset of \[X \cap Y\].
Hence we can conclude \[\left\{ {x:x = 7n - 1,n \in R} \right\}\]belongs to \[X \cap Y = \left\{ {34,......} \right\}\]
So, the correct answer is “Option C”.

Note: Students often get confused with the terms sub-set as the total values that are included in both the sets which is not true. Sub-set contains only the common terms which are present in the given sets.