Question

If $U=\{x:x\in N$ and $2\le x\le 12\}$,
$A=\{x:x$ is an even prime$\}$, $B=\{x:x$ is a factor of $24\}$, then which of the following is not true?

• A. $A-B$ is an empty set
• B. $A-B=B\cap A^{\prime }$
• C. $A^{\prime }-B^{\prime }=B-A$
• D. $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$

Answer 1

Answer: (B)

$U=\{x:x\in N$ and $2\le x\le 12\}$
$\Rightarrow U=\{2,3,4,5,6,7,8,9,10,11,12\}$
$A=\{x:x$ is an even prime$\}$
$\therefore A=\{2\}$
$B=\{x:x$ is a factor of $24\}$
$\therefore B=\{2,3,4,6,8,12\}$
$\therefore A-B=\{2\}-\{2,3,4,6,8,12\}=\varphi$
$\therefore A-B$ is an empty set, is a true statement.
Also, $A-B=B\cap A^{\prime }$ is not correct.
Again, $A^{\prime }=\{3,4,5,6,7,8,9,10,11,12\}$,
$B^{\prime }=\{5,7,9,10,11\}$
$\therefore A^{\prime }-B^{\prime }=\{3,4,6,8,12\}=B-A$, which is true and $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$ is the De Morgan’s law, which is also true.