Question

Consider the following statements :
$1.N\cup (B\cap Z)=(N\cup B)\cap Z$ for any subset $B$ of $R,$ where $N$ is the set of positive integers, $Z$ is the set of integers, $R$ is the set of real numbers.

$2.$ Let $A=\{n\in N:1\le n\le 24,n\text{is a multiple of}3\}.$ There exists no subset $B$ of $N$ such that the number of elements in $A$ is equal to the number of elements in $B.$

Which of the above statements is/are correct?

• A. $1$ only
• B. $2$ only
• C. Both $1$ and $2$
• D. Neither $1$ nor \(2\
  )

Answer 1

The correct option is A $1$ only

$(N\cup B)\cap Z=(N\cap Z)\cup (B\cap Z)$
$=N\cup (B\cap Z)$
$\therefore \left(1\right)$ is correct.

$A=\{3,6,9,12,15,18,21,24\}$
and there exist subset $B$ of $N$ such that the number of elements in $A$ is equal to the number of elements in $B.$
Clearly, we can find $B\subset N$ such that $n\left(A\right)=n\left(B\right)$
$\therefore \left(2\right)$ is not correct.