Let f: (Z Z, ) (Z, +) be the function defined by f(x, y) =

Question:

Let f: (Z × Z, ⊕) → (Z, +) be the function defined by f(x, y) = x - y. [Here (Z × Z, ⊕) is the same group as in Exercise 5, and (Z, +) is the group of integers under ordinary addition.]
(a) Prove that f is a homomorphism onto Z.
(b) Determine all (a, b) ∈ Z × Z with f(a, b) = 0.
(c) Find f-1(7).
(d) If E = [2n|n ∈ Z}, what is f-1 (E)?