basic question about set and functions...

Hi all,

I have been kept annoyed by the following relations about sets and
functions:

suppose f: M -> N is a function. A is contained in N, B is contained in
M. In my notation, "<=" denotes set operation "contained in or equal
to".

1. f(f^(-1)(A)) = A
2. B <= f^(-1)(f(B))

are these two correct? proof?

3. f(f^(-1)(y))= y for all y in N;
4. f^(-1)(f(x)) = x for all x in M;

are thes two correct? proof?

5. A <=B implies f(A) <= f(B) and f^(-1)(A) <= f^(-1)(B)

is this correct? proof?

To me, these are not very regular and difficult to remember and use
them...

Please help me!

.