Question

If among natural numbers $A=\{5,6,7\}$ and $B=\{8,9,10\}$ , then

• A. $A\cap B=$ null
• B. $(A$ $\cup$ $B{)}^{\prime }=A^{\prime }\cap B^{\prime }$
• C. $A$ $\cap$ $B=\{2,3,4\}$
• D. None of these

Answer 1

Answer: (A), (B)

The correct options are
A $A\cap B=$ null
B $(A$ $\cup$ $B{)}^{\prime }=A^{\prime }\cap B^{\prime }$

Since, no elements of A and B are the same.
Therefore, the Intersection is empty.
i.e. $A\cap B=$ null
We know, $U=$ { $N$ }
$A=$ { $5,6,7$ }
$B=$ { $8,9,10$ }
($A\cup B$) $=$ {$5,6,7$) $\cup$ {$8,9,10$}
$=$ {$5,6,7,8,9,10$}
Therefore, ($A\cup B$)'$=$ All natural numbers except $\{5,6,7,8,9,10\}\dots \left(i\right)$
Now, $A^{\prime }=$ All Natural numbers except $\{5,6,7\}$
and $B^{\prime }=$ All Natural numbers except $\{8,9,10\}$
Therefore, $A^{\prime }\cap B^{\prime }=$ All Natural numbers except $\{5,6,7,8,9,10\}\dots \left(ii\right)$
By comparing $\left(i\right)$ and $\left(ii\right)$ we get,
$(A\cup B)=A^{\prime }\cap B^{\prime }$