Question

Let the number of elements of the sets $A$ and $B$ be $p$ and $q$ respectively. Then the number of relations from the set $A$ to the set $B$ is

• A. $2^{p+q}$
• B. $2^{pq}$
• C. $p+q$
• D. $pq$

Answer 1

The correct option is B $2^{pq}$

Given $A$ and $B$ are two sets with number of elements $p$ and $q$ respectively.

The cartesian product of $A$ and $B=A\times B=\left\{\right(a,b):(a\in A)$ and $(b\in B)\}$

Number of elements in $A\times B=|A\times B|=|A|.|B|=pq$

Any relation from $A$ to $B$ is a subset of $A\times B$.

Hence number of relations from $A$ to $B$ is the number of subsets of $A\times B$

$=2^{|A\times B|}=2^{pq}$