Let X be a set, and let A ⊂ X be a subset. Consider the function D : P(X) → P(X) defined by D(B) = A4B. (Recall that the symmetric difference A4B is defined by A4B = (A \ B) ∪ (B \ A).) (a) Is the function D one-to-one for all sets A ⊂ X? If so, prove it; if not, give an example of sets A ⊂ X for which D is not one-to-one. (b) Is the function D onto for all sets A ⊂ X? If so, prove it; if not, give an example of sets A ⊂ X for which D is not one-to-one. (c) Is the function D a bijection for all sets A ⊂ X? If so, prove it; if not, give an example of sets A ⊂ X for which D is not one-to-one. Let n > 1 be an integer. For each x ∈ Z, we can define a function ax : Zn → Zn by f([k]n) = [x + k]n. Prove that ax is always a bijection. (Here Zn is the set of equivalence classes of integers modulo n.)