Question

Given two finite sets $A\&B$ such that $n\left(A\right)=3\&n\left(B\right)=3.$ The total number of relations from $A\to B$ is

Answer 1

Given two sets $A\&B$ such that:
$n\left(A\right)=3\&n\left(B\right)=3$
$\Rightarrow n(A\times B)=3\times 3=9$

Since every subset of $A\times B$ defines a relation from $A\to B,$
Number of relations from $A\to B$ is equal to the number of subsets of $A\times B=2^{n(A\times \text{B)}}=2^9=512$