1. Prove: If X⊆Y then P(X)⊆P(Y).
2. Prove: P(A∩B) = P(A) ∩ P(B) with detail please
3A. Prove that If A⊆B or B⊆A then P(A∪B) = P(A) ∪ P(B)
3B. Prove #3A from the opposite direction, meaning: if P(A∪B) = P(A) ∪ P(B) then A⊆B or B⊆A
4. N is a set of natural numbers N={0,1,2....,} . For every n⊆ N => An = {x∈ N | 0≤ x ≤ n}
Prove or Disprove the following:
a) A0 = Ø
b) ∀n∈N An ⊆ An+1
c) ∃n∈ N An = N
d) ∀n∈N ∀k∈N ∃m∈N |Am - An| = k
e) ∀n∈N ∀m∈N ((Am = {x2 | x∈An}) ↔ (m=n ^ n<2))