Question

Question below

Please Explain each step you did

Answer 1

Let A and B be sets. Prove that if \[P(A∪B)=P(A)∪P(B)\]then \[A \subseteq B\] or \[B \subseteq A\]

Answer 2

P is denoted as a power set

Answer 3

well i worked it out like this: if A⊆B then p(A∪B) = p(A) because A contains all the elements
we do the same for B⊆A: if B⊆A then p(A∪B) = p(B) because B contains all the elements. then we just have to prove that if A!⊆B^B!⊆A then P(A∪B)!=P(A)∪P(B).

a simple proof by contradiction will work here.
assume p(AUB)=P(A)UP(B) when set B has some element x not included in A and
A has some element y that is not included in B. P(AUB) would contain set {x,y} whereas P(A)UP(B) would not contain this set.

Answer 4

hm.. i see thanks a lot for your help!