1) Let F: X -> Y be a function. Justify your answer to each question below by giving either a proof or a counter example.
a) Does f(A - B) = f(A) - f(B), for all A,B subsets of X?
b) Does f(inverse)(C - D) = f(inverse)(C) - f(inverse)(D), for all C,D subset of Y.
Just a not, when I say A - B that is the difference between the sets.
I think I know somewhat how to approach this> I believe I have to prove through the use of set images and that such, but I am stuck. Any help would be great.
There is a quantification error in line 4 of your work.
Look carefully at the definition:
There is an existential quantifier in the definition of the image function.
In the counterexample you can see how that works. In the preimage function the quantification is universal. In fact, image of an intersection is only a subset of the intersection of the images. That is the only case where it fails. Recall that set difference is really an intersection.
And yes A\B is the preferred notation. Even in TeX is ‘setminus’.