What's wrong with this proof

1. Aug 4, 2010

johnnyies

1. The problem statement, all variables and given/known data
Theorem. Suppose x is a real number and x $$\neq$$ 4. If $$\frac{2x - 5}{x - 4}$$ = 3 then x = 7

What's wrong with the following proof of the theorem?

Proof. Suppose x = 7. Then $$\frac{2x - 5}{x - 4}$$ = $$\frac{2(7) - 5}{7 - 4}$$ = $$\frac{9}{3}$$ = 3. Therefore if $$\frac{2x - 5}{x - 4}$$ = 3 then x = 7.

2. Relevant equations

3. The attempt at a solution

2. Aug 4, 2010

johnnyies

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 $$\neg$$Q $$\rightarrow$$ $$\neg$$ P

I'm not sure if that's the reason why the proof is incorrect. The theorem seems fine.

3. Aug 4, 2010

Staff: Mentor

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.

4. Aug 4, 2010

vela

Staff Emeritus
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.

5. Aug 4, 2010

johnnyies

Ya, you're right; that's the question though.

The book told me to refer to a proof strategy in which you have givens, and the statement you want to prove is $$P \rightarrow Q$$ in which you assume P as a given, and prove Q. Which is similar to what you're saying?

6. Aug 4, 2010

Staff: Mentor

That's not only similar to what I'm saying, that's what I'm saying.

7. Aug 4, 2010

johnnyies

I don't still understand how assuming P, then proving Q shows that $$P \rightarrow Q$$ is true?

8. Aug 4, 2010

vela

Staff Emeritus
How does it not?

9. Aug 4, 2010

Staff: Mentor

johnnyies, IMO, you're getting wrapped around the axle fretting about P --> Q and contrapositives and such, when you should just jump in and prove the thing.

You know that x is a real number other than 4, and you know that (2x - 5)/(x - 4) = 3, so what can you conclude from these conditions? By "conclude" I mean what statement can you arrive at by using ordinary mathematical operations.

10. Aug 4, 2010

vela

Staff Emeritus
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.

From a more intuitive viewpoint, for the theorem P→Q to be true, the truth table says if the conditions are met, i.e. P=true, then the conclusion must also hold, i.e. Q=true. If the conditions aren't met, i.e. P=false, then the theorem tells you nothing because the conclusion could hold or not hold, i.e. Q either true or false, and the theorem P→Q would still be true.

On the other hand, if the conditions are met, i.e. P=true and the conclusion doesn't hold, i.e. Q=false, the theorem P→Q must be false.

11. Aug 5, 2010

Char. Limit

Don't think logically, think mathematically. Take your rational equation and solve for x, using only reversibles.

12. Aug 5, 2010

vela

Staff Emeritus
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.

13. Aug 5, 2010

Char. Limit

In that case, the proof begs the question.

14. Aug 5, 2010

Staff: Mentor

I hadn't considered that possibility, but it could very well be the case here.

Looking at the OP's question in that light, the problem was to find the flaw in the proof that P --> Q, where P is the statement, (2x - 5)/(x - 4) = 3, and Q is the statement, x = 7. I am omitting the assumption that x is a real number not equal to 4.

The proof was actually for Q --> P, the converse of the original implication.

15. Aug 6, 2010

HallsofIvy

Staff Emeritus
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.

Last edited: Aug 10, 2010