A relation R on A is a subset of the carthesian product of A with itself, that is: . How many elements does have?
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 then the number of relations on is .
What you are basically doing is trying to find all .
you start out by looking for all then etc...
So you're trying to find all elements of
is the set of all ordered pairs wich has elements. Thus has elements.
(edit: Sorry, I see Plato allready responded)