Let A be the set with n elements. How many elements does the set of relations on A have? (How to calculate tihs?)
I am not sure you're right.
We don't talk about sets but about ordered pairs:
now I have to found all relation: I think that I have to do something like this
R_1={(a,b)} there are n*n different options
R_2={(a,b),(c,d)} I think there are (n*n)*((n-1)*n + (n-1)) = n^2*(n-1)*(n+1)
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(this is something about combinatorics-I think so)
The first reply is absolutely correct.
As for you statement “We don't talk about sets but about ordered pairs” that is incorrect.
Almost all of basic ideas in mathematics are about sets.
An ordered pair is a set of sets.
A relation on a set is a subset of the cross product of the set with itself as was stated.
If we have a finite set $\displaystyle A$ then the number of relations on $\displaystyle A$ is $\displaystyle 2^{|A|^2}$.
What you are basically doing is trying to find all $\displaystyle R\subset A\times A$.
you start out by looking for all $\displaystyle R_1\subset A\times A$ then $\displaystyle R_2 \subset A\times A$ etc...
So you're trying to find all elements of $\displaystyle \mathcal{P}(A\times A)$
$\displaystyle A\times A$ is the set of all ordered pairs $\displaystyle (a,b)$ wich has $\displaystyle n^2$ elements. Thus $\displaystyle \mathcal{P}(A\times A)$ has $\displaystyle 2^{n^2}$ elements.
(edit: Sorry, I see Plato allready responded)