Prove or disprove: if A, B, and C are sets where A-C = B-C, then A = B. This is my first disproof and I have a couple of questions. 


*

*Can you just disprove with a counterexample? 

*The question doesn't say "for all", does that mean I automatically imply that the claim is "true" for every set?

*If someone can critique my proof writing and see if there's a better way of writing this proof?
My working: 
A = {0,1,2,3,4}
B = {0,1,2,3,4,5,6,7,8}
A $\neq$ B
C = {4,5,6,7,8}
A-C = {0,1,2,3}
B-C = {0,1,2,3}
A = B
Proof: 
We will show that claim is false. By negating the initial claim, we can rewrite it as A-C = B-C ^ A $\neq$ B. 
Let us consider a set A = {x $\in \mathbb N$ | x $\le$ 5}, B = {x $\in \mathbb N$ | x $\le$ 8} and C = {x $\in \mathbb N$ | x $\ge$ 5 $\cap$ x $\le$ 8}. 
By performing A - C we result with a set where A-C={x $\in \mathbb N$ | x $\le$ 4}. Similarly consider the set B-C = {x $\in \mathbb N$ | x $\le$ 4}.
This leads to the conclusion that the claim is false, since the we have proved the negation equivalent as true. 
 A: *

*It can be disproved with a counterexample.

*We understand the sets A, B and C as arbitrary sets, so if true, the statement holds for all sets A, B and C.

*What you need to do is to give two different sets such that when you take the difference for each of those with a particular set, the remaining sets are the same set, so the statement is false.


e.g:
$A = (-\infty, 0]$, $B= (-\infty, 1]$, $C= [-2, 1]$ 
So $A-C = (-\infty, -2)$, $B-C = (-\infty, -2)$ but $A\neq B$
A: Too long for a comment

Can you just disprove by a counterexample?

Yes, if you found a counterexample, the claim is false.

The question doesn't say "for all", does that mean I automatically imply that the claim is "true" for every set?

The claim is an implication $P\implies Q$ whenever the premice are true, the conclusion must also be true. We don't need "for all".

If someone can critique my proof writing and see if there's a better way of writing this proof?

Your proof is fine. Your three sets respect the hypothesis, but not the conclusion, therefore the claim is false.
