Question

Assertion (A) Let $A=\{1,2\}$ and $B=\{3,4\}$. Then, number of relations from $A$ to $B$ is 16 .
Reason (R) If $n(A)=p$ and $n(B)=q$, then number of relations is $2^{p q}$.

• A. Both $A$ and $R$ are correct; $R$ is the correct explanation of $A$
• B. Both $A$ and $R$ are correct; $R$ is not the correct explanation of $A$
• C. $A$ is correct; $R$ is incorrect
• D. $R$ is correct; $A$ is inco rect

Answer 1

Answer: (A)

The total number of relation that can be defined from a set $A$ to a set $B$ is the number of possible subset of $A \times B$. If $n(A)=p$ and $n(B)=q$, then $n(A \times B)=p q$ and the total number of relation is $2^{p q}$.
Given, $ A=\{1,2\}$ and $B=\{3,4\}$
$\therefore A \times B=\{(1,3),(1,4),(2,3),(2,4)\}$
Since, $n(A \times B)=4$, the number of subsets of $A \times B$ is $2^4$.
Therefore, the number of relation from $A$ to $B$ will be $2^4=16$.
Note $A$ relation $R$ from $A$ to $A$ is also stated as a relation on $A$.