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.
Best Answer
Too long for a comment
Yes, if you found a counterexample, the claim is false.
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".
Your proof is fine. Your three sets respect the hypothesis, but not the conclusion, therefore the claim is false.