Is the number of relations on $A \times B$ the same as the number of relations from $A$ to $B$?
Can anybody clear this doubt with some examples? In my notes, I have written the number of relations on $A \times B$ as $2^{(mn)^2}$ and number of relations from $A$ to $B$ as $2^{mn}$. But some friends are arguing both are same and answer is $2^{mn}$.
I think you are right.
A relation from $A$ to $B$ is a subset of $A\times B$.
A (binary) relation on a set $X$ is a relation from $X$ to $X$ hence is a subset of $X\times X$.
So a (binary) relation on set $A\times B$ is a subset of $(A\times B)\times(A\times B)$.
See here for example.
I can imagine though that sometimes mathematicians are kind of sloppy by the use of this terminology. So do not put too much trust in it and inform.