# AunionB=AunionC, B=C

1. Nov 17, 2008

### kathrynag

1. The problem statement, all variables and given/known data
I need to decide whether the theorem is correct and decide where the proof fails.
AunionB=AunionC, B=C

2. Relevant equations

3. The attempt at a solution
We will prove by contradiction. Suppose that A$$\cup$$B and A$$\cup$$C are not equal. Then there is some object x that is in one and not the other. We proceed by looking at 2 cases:
First look at the case where x$$\in$$A$$\cup$$B and x$$\notin$$A$$\cup$$C. Then x$$\notin$$A. So x$$\in$$B. Also x$$\notin$$C. Therefore x is in B and not in C, which contradicts the condition B=C.

I thought the theorme was correct, but I can't find where the proof goes wrong.

2. Nov 17, 2008

### HallsofIvy

Staff Emeritus
That's NOT the way proof by contradiction works! To prove "if X then Y" by contradiction, you assume Y is not true. Here your theorem is "if $$A\cup B= A\cup C$$ then B= C. Proof by contradiction would start "suppose B is not equal to C".

3. Nov 17, 2008

### kathrynag

Oh yeah I forgot about that.

4. Nov 18, 2008

### crespo

I think it's not true. You take A={1,2,3}. B={1,0}, C={2,0}. Then B not = C but A union B= A union C