Prove: If A-B=B-A, then A=B

[toughcookie723]

1.Suppose A, B are two sets. Prove the statement : If A-B=B-A, then A=B.

A-B=B-A
A+A=B+B

A U A= B U B
By definition of sets, since there can only be one of the same element in a set A U A =A. Therefore,

A U A=A
B U B=B

So A=B. --> iS THIS A CORRECT PROOF?

2. Prove: If A intersection B= A U B, then A=B. (Hint: see #1).

(Well, here I am sorta stuck. I tried working backwards but didn't get far. Can you please help?!!!)

A INT B=A u B

(A u A)INT B= (B u B) u A
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A-B=B-A
B=A

(all the dots after the second statement are the parts I am missing and no idea how to get to the end or for that matter up to the top from the bottom)

PLEASE help! :-?

 

[pka]

1.Suppose A, B are two sets. Prove the statement : If A-B=B-A, then A=B.
A-B=B-A
A+A=B+B

Where did that come from? It is wrong!
\(\displaystyle A-B=A\cap B^c\) where \(\displaystyle B^c\) is the complement of \(\displaystyle B\).

Now if \(\displaystyle A\ne B\) then \(\displaystyle \left( {\exists x} \right)\left[ {x \in A\cap B^c\text{ or }x\in B\cap A^c} \right]\)

From the given, is that possible?