the cardinal number of the set of the relations

Question

What is the cardinal number of the set of the relations from N to N *N?

I think is similar to all subset so 2 ^ א0 but it's not exactly and I don't know how to take it from there. I have a test on an hour. please help..

Answer 1

Well, the number of binary relations from a set $A$ with $n$ elements to a set $B$ with $m$ elements is given by $2^{n\times m}$.

If you want the number of relations from $A$ to $A\times A$ simply use the above mentioned rule by taking $B=A\times A$. Thus, the set of all relations would yield a cardinality of $2^{n^3}$ where $n$ is the cardinality of $A$.

PS - you may refer to this https://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class21b.pdf for more information.