How Do You Prove Power Set Equations for Sets A and B?

[JasonJo]

How in the heck do i prove these:

Prove whether the following equations are true for all sets. For each one that's not always true, try to prove that one side is a subset of the other, and give a counterexample to the other direction. If neither side must be a subset of the other, give a counterexample to both directions:

let P(A) denote the power set of A

a) P(A U B) = P(A) U P(B)
b) P(A-B) = P(A) - P(B)

 

[0rthodontist]

a. is not true. Think about it, try an example or two. The subset goes from right to left. To prove things like this, you assume x is an element of P(A) U P(B) and show that x must be an element of P(A U B).

b. is also not true, the subset goes from left to right.

 

[JasonJo]

for part (b) i don't think either side is a subset of each other because:

let A = {1,2}
B = {1,3}
A-B = {2}

P(A) = {0, 1, 2, {1,2}}
P(B) = {0, 1, 3, {1,3}}
P(A)-P(B) = {2, {1,2}}
P(A-B) = {0, 2}

they aren't equal and one isn't a subset of another.

 

[0rthodontist]

Ah, you're right.