Proving AunionB=AunionC, B=C Theorem

[kathrynag]

Homework Statement

I need to decide whether the theorem is correct and decide where the proof fails.
AunionB=AunionC, B=C

Homework Equations

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.

 

[HallsofIvy]

Homework Statement

I need to decide whether the theorem is correct and decide where the proof fails.
AunionB=AunionC, B=C

Homework Equations

The Attempt at a Solution

We will prove by contradiction. Suppose that A$$\cup$$B and A$$\cup$$C are not equal.

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".

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.

 

[kathrynag]

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".

Oh yeah I forgot about that.

 

[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