Prove that if A and B are sets, then (A - B) U B = A U B

[leo255]

Homework Statement

Prove that if A and B are sets, then (A - B) U B = A U B

I think I might be missing a few steps here.

Homework Equations

The Attempt at a Solution

(A - B) U B =

1. (A ^ ~B) U B =

2. (A ^ ~B) U (A ^ B) =

3. A U B

 

[Buzz Bloom]

Hi Leo:

I think that whether a step is "missing" or not depends on what your teacher expects.
Hint:
(W ^ X) U (Y ^ Z) = ((W U Y) ^ (X U Y)) U ((W U Z) ^ (X U Z))

 

[S.G. Janssens]

Issue with step 1 to 2: When I'm in $B$, but not in $A$, I'm in set (1) but not in set (2).
Issue with step 2 to 3: Set (2) is simply equal to $A$, which may be smaller than set (3).

I recommend you do not try to "rewrite" the sets, but instead prove two inclusions: First take an arbitrary $x \in (A - B) \cup B$ and argue step by step why it is then in $A \cup B$. Then take an arbitrary $x \in A \cup B$ and argue why it is in $(A - B) \cup B$.