The discussion revolves around a theorem involving a real number x, specifically addressing the validity of a proof that claims if \(\frac{2x - 5}{x - 4} = 3\), then \(x = 7\). Participants are examining the logical structure of the proof and the implications of assuming certain values for x.
The conversation is ongoing, with participants providing insights into the logical flaws of the original proof and exploring the implications of the statements involved. Some guidance has been offered regarding the structure of logical proofs, but there is no explicit consensus on the resolution of the misunderstanding.
There is a focus on the logical structure of proofs, particularly the distinction between direct implications and converses. Participants are also considering the assumptions made in the theorem and the proof strategy outlined in the original post.
johnnyies said:Therefore if \frac{2x - 5}{x - 4} = 3
then x = 7.
Yeah, that's the problem with the proof. Assuming what you want to prove and showing the assumption leads to a true statement doesn't tell you anything about whether the initial assumption was correct because (F→T)=T.johnnyies said:Crap. I accidentally clicked submit instead of preview post.
Attempt at solution:
Our two givens for this proof is that x is a real numberand x is not 4, and the statement is in an if P, then Q: P \rightarrow Q. However, the proof is in the form of the converse, if Q, then P: Q \rightarrow P, which is not equal to the statement. The proof should be in form of P \rightarrow Q or its contrapositive \negQ \rightarrow \neg P
I'm not sure if that's the reason why the proof is incorrect. The theorem seems fine.
Mark44 said:In your proof in your first post, you are assuming that x = 7, and then concluding that x = 7. This is not very interesting.
The second thing to point out is that you say
That (the part after if) is precisely what you need to prove, and you have completely glossed over it.
This is a very simple proof, so you don't need to (and shouldn't) work with the contrapositive. Just start from the three given conditions, a very important one of which is (2x - 5)/(x - 4) = 3, and conclude that x = 7.
When you assume P, you mean statement P is true. When you prove Q, you mean statement Q is true. From the truth table for P→Q, you can see P→Q is true.johnnyies said:I don't still understand how assuming P, then proving Q shows that <br /> P \rightarrow Q <br /> is true?
I hadn't considered that possibility, but it could very well be the case here.vela said:I don't think the OP's question is about how to prove this specific theorem. It's about the logical structure of proofs, what's acceptable and what's not. He or she was given an example proof and asked to determine whether and why it was valid or not. I'm guessing this is from a "how to write proofs" course.
Because that's exactly what P\rightarrow Q means! "If P is true then Q is true". Proving that any time P is true (assuming P), Q is also true is exactly what proving P\rightarrow Q means.johnnyies said:I don't still understand how assuming P, then proving Q shows that <br /> P \rightarrow Q <br /> is true?