Question

Let $N$ denote the set of natural numbers and $A=\{{{n}^{2}}:n\in N\}$ and $B=\{{{n}^{3}}:n\in N\}$, which one of the following is incorrect?
A. $A\cup B=N$
B. The complement of $(A\cup B)$ is an infinite set
C. $A\cap B$ must be a finite set
D. $A\cap B$ must be a proper subset of $\{{{m}^{6}}:m\in N\}$

Answer 1

Hint: In this question, we are to find the incorrect statement in the above-given statements. Since the sets are defined, we can find the given statements

Formula Used:Set: A collection of objects in which it is possible to decide whether a given object belongs to the collection or not is said to be a set. Those objects are nothing but the elements in the set.
Sets are represented in two ways: The roaster method and the Set builder form
Roaster method:
The set of vowels – \[\left\{ a,\text{ }e,\text{ }i,\text{ }o,\text{ }u \right\}\]
The set of natural numbers – \[\left\{ 1,\text{ }2,\text{ }3,\text{ }\ldots. \right\}\]
Set builder form:
The set of vowels – \[\left\{ x:x\text{ }is\text{ }a\text{ }vowel\text{ }in\text{ }English\text{ }alphabet \right\}\]
The set of natural numbers – \[\left\{ x:x\text{ }is\text{ }a\text{ }natural\text{ }number \right\}\]
Some of the important mathematical sets:
$N$ - the set of natural numbers - $N=\{1,2,3...\}$
$Z$- the set of integers - $Z=\{0,\pm 1,\pm 2,\pm 3,...\}$



Complete step by step solution:The given sets are:
$A=\{{{n}^{2}}:n\in N\}$ and $B=\{{{n}^{3}}:n\in N\}$
Then, we can write,
$\begin{align}
  & A=\{1,4,9,16,...\} \\
 & B=\{1,8,27,64,...\} \\
\end{align}$
So,
$A\cap B=\{1\}$ is a finite set
Thus, statement (3) is correct.
$A\subset N;B\subset N\Leftrightarrow A\cap B\subset N$
The set $\{{{m}^{6}}:m\in N\}$ is also a subset of $N$ i.e.,
$\{{{m}^{6}}:m\in N\}=\{1,64,...\}\subset N$
Therefore, $A\cap B$ is a proper subset of $\{{{m}^{6}}:m\in N\}$.
So, statement (4) is also true.
$A\cup B=\{1,4,8,9,16,25,27,...\}$ is an infinite set and $A\cup B\ne N$
So, statement (1) is false/incorrect.
Since $A\cup B$ is an infinite set, the complement of $A\cup B$ is also an infinite set.
So, statement (2) is correct.
Thus, from the given statements, statement (1), $A\cup B=N$ is incorrect.



Option ‘A’ is correct


Note: Here, the sets are subset of $N$, so their union and intersection also subset of $N$. But they are not the same means they are proper subsets of $N$. By this kind of analysis, the above statements are verified and the incorrect statement is found.