[Math] Prove or disprove: if A, B, and C are sets where A-C = B-C, then A = B.

discrete mathematicssolution-verification

This is my first disproof and I have a couple of questions.

  1. Can you just disprove with a counterexample?

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

  3. 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

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.

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