Proving Equivalency of 1-1 Functions: Intro to Real Analysis

[mmathew23]

Let f: X--> Y where X and Y are arbitrary sets. Show the following are equivalent.

a. F is 1-1 on X
b. f(A/B)=f(A)/f(B) for all subsets A and B of X
c. f^-1 f(E) = E for all E that is a subset of X
d. f(A intersect B) = f(A) union f(B) for all A,B that is a subset of X

I know that in order to f to be 1-1 on X one value of x corresponds to one value of Y. For example f(x1)=a and x1 does not equal x2 so therefore f(x2) can't equal a. Other than that I don't know where to start to show that these all imply the other. I am also having a hard time understanding what b and d even mean. Any help would be great.

 

[Kummer]

Try to write out in words what these means.
For example f(A/B) = { f(x) | x in A but x not in B}.
This makes it much easier.