Let x be a set and px its powerset define a function f px

Question

Q1. Consider the sets X = {4m + 1|m ∈ Z} and Y = {2n - 1|n ∈ Z}.

(a) Prove that X ⊆ Y.

(b) Is Y ⊆ X? If yes, prove it. If not, find an element of Y that's not an element of X.

Q2. Let X be a set, and P(X) its powerset. Define a function f : P(X) → P(X) by f(A) = X\A.

(a) Consider the case X = {1, 2, 3}. Evaluate f(A) for every A ∈ P(X).

(b) For X any set, prove that f is a bijection.