# Thread: Proof involving power set and union

1. ## Proof involving power set and union

Hello guys,
I need help on this question.

Let A and B be sets. Let A = {Sk | k ∈ I } be a family of sets with I not equal to an empty set. Prove the statement.
If P(A U B) ⊆ P(A) U P(B) then A ⊆ B or B ⊆A

What I have in mind right now::
Assume
P(A U B) ⊆ P(A) U P(B)
Let X
P(A U B) which X ∈ A or X ∈ B
demonstrate 2 the cases.
Case 1:
X ∈ A
From the assumption Show A ⊆ B
Case 2: X
∈ B
From the assumptionShow B ⊆A
therefore A ⊆ B or B ⊆A

I might be wrong, but that is all i have right now and
i don't know how to express the proposition
P(A) U P(B) or how to use it.
if anyone can explain it, that would be awesome.

Any help would be appreciated.
Thank you.

2. ## Re: Proof involving power set and union

I would prove the contrapositive, show that if there is x in A-B, AND y in B-A, that then there is an element of P(AUB) that does not lie in P(A)UP(B).

Hint: consider the set {x,y}.

3. ## Re: Proof involving power set and union

Thank you. I got it.