Does P(A) U P(B) Subset of P(A U B)?

[pinkyjoshi65]

Homework Statement

Is this statement true?

For all sets A,B contained in a universe U, P(A) U P(B) is a subset of P(A U B) if and only if A is a subset of B or B is a subset of A

 

[Mark44]

What is P?

 

[Ja4Coltrane]

It's sort of pf policy that you show some attempt at a problem before we try to help.

you should try looking at various disjoint sets. For instance, take U = set of all integers, take A={0},B={1}.

 

[Ja4Coltrane]

Mark, P(A) is probably the power set of A.

 

[pinkyjoshi65]

P is the power set.

 

[pinkyjoshi65]

this is just a part of a question. I did try doing it. Here is the actualy question:

for sets A and B, P(A intersection B) = P(A) intersection P(B). However,
the same property does not hold for unions. To fully investigate the corresponding
property for unions, do the following exercise:
Let A and B be sets contained in a universal set U.
(a) Prove that P(A) U P(B) is a subset P(A U B).
(b) Give examples of sets A and B, for which P(A) U P(B) is not equal to P(A U B).
(c) Under what conditions on A and B will P(A) U P(B) = P(A U B)?
State your answer in the form of a theorem: i.e.
”Theorem
For all sets A and B contained in a universe U, P(A) U P(B) = P(A U B)
if and only if ... ”
(d) Prove your theorem from part (c).

For a) I have the following: Assume x belongs to P(A) U P(B)
Hence X is a subset of A or x is a subset of B
Let Z belong to X
Hence Z belongs to A or B
hence Z belongs to A U B
Hence x is a subset of A U B
Hence x belongs to P(A UB)

 

[pinkyjoshi65]

For part b) An example is A= {1,2,3}, B= {2,3,4}
A u B ={ 1,2,3,4}
Let x= {1,4} X is a subset of A U B but not a subset of A or B