power of the set of all relations over a finite set

Question

I need to find the power of the set of all relations over a set with $n$ members. The set of all relations over a set with $n$ members is $n^2$. The power of such a set is $2^{n^2}$. For example, if we had a set $\{1,2,3\}$ the power of the set of relations over the set would be $2^9$. Is is correct?

Answer 1

Your calculation is correct, and I’m pretty sure that your reasoning is correct as well. However, your second sentence isn’t correct as written, and judging by the rest of your answer, it doesn’t say what you actually meant. It and the following sentence should read something like this:

The relations over a set $A$ with $n$ members are the subsets of $A\times A$, and $A\times A$ has $n^2$ members. Its power set therefore has $2^{n^2}$ members.