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.

IfP(A U B) ⊆ P(A) U P(B)thenA ⊆ 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.