My name is Michael, I'm a freshman CS student and I'm stuck with this problem. I could really use your help:
Let T be a non empty set, and let A and B two sets belonging to P(T), the set of partitions of T ( whatever x belongs to P(T), the complement of x = T - x )
Now, let f be a function, f defined on P(T) with values in the direct product P(A) x P(B), f(x) = ( x intersected with A, x intersected with B )
I must prove that if f is Injective if and only if the Union of A and B is T.
There are two more points to the problem, but I'm hoping if I understand how to do this, I can solve the other two myself. I tried using the Cantor-Bernstein-Schroeder theorem, but I didn't got me anywhere. I could use any help.