Let $A$ and $B$ be finite sets, $|A| = m$ and $|B| = n$. How many binary relations are there from $A$ to $B$?

As far I read from various resources, binary relations is nothing but just a typical (total) function. Then, does it mean that such possibilities are equal to $n^m$, or not?

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    $\begingroup$ No, absolutely not: most relations are not functions. A relation from $A$ to $B$ is simply a subset of $A\times B$; how many of those are there? $\endgroup$ – Brian M. Scott Nov 14 '20 at 0:07
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    $\begingroup$ There are $2^{mn}$ number of subsets of Cartesian product, but I can not analyze how can we build such binary relations using these Cartesian product subsets $\endgroup$ – Anar Rzayev Nov 14 '20 at 0:07
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    $\begingroup$ Yes, that’s right, and each of them is a relation from $A$ to $B$. $\endgroup$ – Brian M. Scott Nov 14 '20 at 0:08
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    $\begingroup$ The question that you’ve written justs asks for the number of binary relations from $A$ to $B$, so it simply wants a number; the point is to find out whether you know that the relations from $A$ to $B$ are just the subsets of $A\times B$ and whether you know how to calculate how many subsets there are. $\endgroup$ – Brian M. Scott Nov 14 '20 at 0:15
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    $\begingroup$ You’re welcome! $\endgroup$ – Brian M. Scott Nov 14 '20 at 0:21

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