How to disprove a theorem

Question

I have a question that says,

Explain how to disprove a theorem of the logical form "$\forall x \in A, P(x)$". Write the logical form of the statement you want to prove.

So disprove a theorem, wouldn't you just find a counterexample because it doesn't matter what logical form the equation is in right? Then for the logical statement you want to prove, would it just be the negation of "$\forall x \in A,P(x)$"?

Yes, you just need to demonstrate $\exists x_0\in A$ such that $\neg P(x_0)$. — yurnero, May 23 '16 at 17:23
To disprove that all swans are white, you find a swan that is not white. $\qquad$ — Michael Hardy, May 23 '16 at 17:59

score 1 · Answer 1 · answered May 23 '16 at 17:23

1

You are right in saying that you should find a counterexample. The logical form of this counterexample would state that $\exists x \in A, \neg P(x)$. (When you're dealing with logic, you have to be careful how you perform the 'negation'.)

answered May 23 '16 at 17:23

shardulc

4,562

score 0 · Answer 2 · answered May 23 '16 at 17:24

0

Yes you are correct. One counterexample is enough to disprove a theorem.

You can check whether it is a counterexample by taking all conditions for the theorem and then negating the proposition.

So if you have for example $\forall_{x \in A} : P(x)$, where $P$ is your proposition. Then negating this turns into $\exists_{x \in A} : \neg P(x)$, which disproves the theorem.

answered May 23 '16 at 17:24

Tim Huijgens

342

1

To disprove such a theorem, one counterexample is enough. But that's not so for other types of theorems, i.e. theorems whose form isn't "for all ...". – zipirovich May 23 '16 at 17:29

How to disprove a theorem

2 Answers2