Question

Let $A$ and $B$ be two smallest sets such that $A\cup \left\{1\right\}=\{1,2,3,4\}$ and $B\cup \left\{5\right\}=\{4,5,6,7,8\}.$ If $P=A-B$ and $Q=B-A,$ then the number of relations from $P$ to $Q$ is

• A. $2^9$
• B. $2^{12}$
• C. $2^{10}$
• D. $2^6$

Answer 1

The correct option is D $2^6$

$A\cup \left\{1\right\}=\{1,2,3,4\}$
$\Rightarrow A=\{2,3,4\}$ (smallest possible $A$)
$B\cup \left\{5\right\}=\{4,5,6,7,8\}$
$\Rightarrow B=\{4,6,7,8\}$ (smallest possible $B$)
$P=A-B=\{2,3\}Q=B-A=\{6,7,8\}$
Hence, number of relations from $P$ to $Q$ is, $2^{n\left(P\right)\cdot n\left(Q\right)}=2^6$