# Not sure why this proof was marked wrong, can you?

1. Oct 13, 2006

### mr_coffee

Hello everyone i'm reviewing for the exam and I got 0 points on this proof and it seems like a nice exam question. So i want to make sure I have the correct proof to study.

I don't see how this is wrong when he did a proof such as this one, in almost the same way as mine:

Any help with correcting my above proof so I may study off it would be great!

Note: he did not correct these, the TA did.

Last edited: Oct 13, 2006
2. Oct 13, 2006

### quasar987

Sorry for replying while not providing an iota of information relative to your question but what does TA stands for?

3. Oct 13, 2006

### mr_coffee

quite alright, Teachers Assistant
Here was the grading instructions to the TA:

Grading instructions: An answer is worth 10 points if it gives a speci c counterexample, such as \ 0=p2 = 0,
which is rational." In such a counterexample, the numerator has to be zero, and the denominator has to be
a number that the textbook proved was irrational, such as p2. You may also count as correct an answer that
consists of a general statement such as, \If r = 0 then r=s = 0, and this is rational even when s is irrational."
Note: The idea here is that students who don't happen to think of the case r = 0 will start to write out a
proof that r=s is irrational, and will reach a point in the proof where the de nition of rational requires that the
denominator be a nonzero integer. At that point, it will not be possible to show that the denominator is nonzero
unless r is nonzero, which it doesn't have to be. That should cause students to realize that r could be zero, and
that, when r = 0, r=s is not irrational. So students should in the end come up with a counterexample even if
they did not think of one initially.
If a student writes a false \proof" that r=s has to be irrational, simply score the answer as zero points. (This can
happen when a student tries to prove a number is rational without bothering to verify the part of the de nition
that requires the denominator to be nonzero. Each proof in the textbook and in class that showed a number was
rational by using the de nition was careful to check this condition, so the mistake is probably not momentary
carelessness, but is more likely the result of someone not yet having learned to write proofs carefully enough to
avoid logical errors.)

4. Oct 14, 2006

### Galileo

This was more or less what I wanted to post. A simple counterexample to the claim, like 0/e=0 would have given you 10 points.
Such things are hard lessons we've all had to deal with.

5. Oct 14, 2006

### mr_coffee

How would I disprove with a counter example correctly so on an exam i get all points? Is it simply stating a counter example?

Proof by counter example:
0/sqrt(2) = 0, 0 is rational, sqrt(2) is irrational but the result is rational, which is rational therefore contradicts the claim.

Or with a counter example, do i have to take the negation of the statement and do a proper proof? I don't see how this is possible if your just giving a simple counter example but I want to make sure.

Thanks for the responce.

Last edited: Oct 14, 2006
6. Oct 15, 2006

### Galileo

One counterexample is enough to disprove any claim. It IS a proper proof that the claim is false.

Read your TA's note on this. You can use a specific counterexample, like the one you posted with s=sqrt(2), but you must then have proven (from class or in the textbook) that sqrt(2) is indeed irrational. Or you can say 0/s=0 for ANY s(/=0), rational or irrational.

7. Oct 15, 2006

### JasonRox

Yeah, but you should be able to prove that it is irrational if r is in the set Q/{0}.

The proof you wrote was long for something so trivial.

You even wrote if s=/=0, when s is irrational. How can s=0 and be irrational?

8. Oct 15, 2006

### mr_coffee

yeah your right....im not sure what other example I would use other than 0. Becuase in the TA's grading notes it says, They must use 0 as the numerator, and they must use sqrt(2) because the text book proved sqrt(2) is irrational, i'm not sure why he is making us use 0 as the numerator but he is. \

Thanks for the responces!

9. Oct 15, 2006

### HallsofIvy

Staff Emeritus
The point is that 0/s is the only case in which r/s is rational with s irrational.

You can prove (in fact, you basically did) : if s is irrational and r is a non-zero rational number then r/s is irrational: Suppose, to the contrary, x= r/s is rational. Then sx= r and since s is non-zero, we can divide by it, so s= r/x. But the rational numbers are closed under division (dividing by a non-zero number, of course), contradicting the fact that s is irrational/

10. Oct 15, 2006

### mr_coffee

ahhh thanks for the clarification Ivy!
i get it now!