Is it technically incorrect to write proofs forward?

Question

A question on an assignment was similar to prove: $$2a^2-7ab+2b^2 \geq -3ab.$$ and my proof was: $$2a^2-4ab+2b^2\geq0$$ $$a^2-2ab+b^2\geq0$$ $$(a-b)^2\geq0$$ which is true.

However, my professor marked this as incorrect and the "correct" way to do it was:

Starting from $$(a-b)^2\geq0$$ we have: $$a^2-2ab+b^2\geq0$$ $$2a^2-4ab+2b^2\geq0$$ $$2a^2-7ab+2b^2 \geq -3ab.$$

His point was that if we start with a false statement we can also reduce it to a true statement (like $-5 =5$ we can square for $25 = 25$). I argued however that you can go back and forth between my operations (if and only if, which doesn't work for the $-5 =5$ example). He still didn't give me the marks for it. Which leads me to my questions:

1. Is my proof equally valid?

2. Do real mathematicians all write one way or the other when writing in a paper?

Answer 1

There are two ways to interpret your argument (which hardly counts as a proof if this is exactly what you have written in your assignment).

First Way

You actually meant to write, \begin{align}2a^2-7ab+2b^2 \geq -3ab&\implies 2a^2-4ab+2b^2\geq0\\&\implies a^2-2ab+b^2\ge 0\\&\implies (a-b)^2\ge0\end{align}

Second Way

You actually meant to write, \begin{align}2a^2-7ab+2b^2 \geq -3ab&\iff 2a^2-4ab+2b^2\geq0\\&\iff a^2-2ab+b^2\ge 0\\&\iff (a-b)^2\ge0\end{align}

In the second case your argument is correct but in the first case it is not (as your professor has elaborated via an example).

Now the answer to your questions,

1. If you didn't say explicitly in your assignment in which way your argument is to be interpreted and then if your teacher interprets your argument in the first way, you can't blame him/her for not giving you the corresponding marks of the question simply because it was you who failed to be explicit )and the first way of interpreting your argument indeed shows a misunderstanding of the working of $\implies$). So, if I were in the position of your professor, I would give you no marks.

2. I don't know what you mean by "one way or the other" here. The best way to answer this question will be to read the papers of some real mathematicians. However, when real mathematicians write a paper it is in general clear what they are assuming to be true and how the steps lead to the conclusion (which is not the case in your argument as I have explained above).

Answer 2

Your professor is correct. Whether or not it's clear "what you mean," in my teaching experience this mistake exposes an extremely common misconception about how proofs work. In a proof, you're trying to communicate how a logical conclusion follows from one or more logical premises. A proof is never just a sequence of assertions: it must communicate the logical relationships between those assertions. Whenever we get lazy and just write the assertions, the implicit logical relationship is that each one follows as a logical consequence from the assertions written before it. That's not what you want here, though: the premise needs to be something you know, and the conclusion needs to be what you're trying to show.

So, whenever you're tempted to write one assertion after another, think to yourself: is this assertion logically equivalent to the one before it? Does it imply the one before it? Is it implied by the one before it? The sooner you get in the habit of putting those relationships in writing, the better. As commenters have said, it can be as simple as putting little arrows between your statements that point from logical premise to logical conclusion.

(In "real" mathematical writing, you can write premises before conclusions or conclusions before premises, but it's always clear which is which. It should be that way in your writing, too.)

Answer 3

What you wrote, unless you made the equivalence marks explicit, is:

\begin{align}2a^2-7ab+2b^2 \geq -3ab\implies ...\implies (a-b)^2\ge0\end{align}

In other words you've shown that if the left side is true, then it follows that the right side is too. Knowing that the right side is true doesn't allow you to say anything about the left side. (What you can say, though, is that if the right side is false, then it'll follow that the left side is also false.)

What your professor wrote, by contrast, and assuming no explicit equivalence marks either, is:

\begin{align}(a-b)^2\ge0\implies ...\implies 2a^2-7ab+2b^2 \geq -3ab\end{align}

And thus, because the left side is true, it follows by implication that the right side is too.

1. Should I have received marks for this question?

I don't teach, but FWIW I wouldn't have given you any marks either.

2. Do real mathematicians all write one way or the other when writing in a paper?

It's not about writing it one way or the other, it's about getting the implications in the correct order.

Answer 4

I'm going to emphasize something that I don't see in the other answers. The point of a proof is ultimately to be a persuasive argument to the reader (Richard Lipton has written on this at least once). The first statement should be a self-evidently true fact (or an assumption of the hypothesis). It would be weird for the reader to be in the dark as to the truthfulness of all the statements in the proof until the end of the reading.

I will use as an example a fragment from the Sherlock Holmes story, "Adventure of the Dancing Men". In one scene Holmes astonishes Watson by deducing from a small indentation on his finger that he will not be investing in South African securities. He reasons aloud:

1. You had chalk between your left finger and thumb when you returned from the club last night.
2. You put chalk there when you play billiards, to steady the cue.
3. You never play billiards except with Thurston.
4. You told me, four weeks ago, that Thurston had an option on some South African property which would expire in a month, and which he desired you to share with him.
5. Your check book is locked in my drawer, and you have not asked for the key.
6. You do not propose to invest your money in this manner.

Consider if this series of statements were reversed:

1. You do not propose to invest your money in this manner.
2. Your check book is locked in my drawer, and you have not asked for the key.
3. You told me, four weeks ago, that Thurston had an option on some South African property which would expire in a month, and which he desired you to share with him.
4. You never play billiards except with Thurston.
5. You put chalk there when you play billiards, to steady the cue.
6. You had chalk between your left finger and thumb when you returned from the club last night.

While not entirely incoherent, the reversed sequence seems noticeably harder to follow the connections than the original. This is similar to what you were doing in your proposed proof. While the convention is largely a stylistic one, it is indeed mostly followed by mathematicians -- but moreover, presenting in the forward direction is widely followed by any good writer or speaker, who wants to make their thought as transparent and persuasive as possible, to any reader or listener.

Notice that in the original text, Holmes starts with something that is undeniably verifiable by everyone present (the fact that Watson has an indentation on his finger), and gets him to agree to that first. It is well known psychologically that saying "yes" primes to say another "yes" in sequence (in sales, this is called the Yes-Set Close). While we should use such techniques responsibly, as persuasive writers, we should not fail to use such tools if they are available.

Answer 5

I'll start by answering your second question:

2. Do real mathematicians all write one way or the other when writing in a paper?

If you take a random paper in mathematics (e.g. from ArXiv) and see the proof, there are very good chances that you won't see either stuff like

\begin{align} (a-b)^2& \geq0 \\ a^2-2ab+b^2 &\geq0\\ 2a^2-4ab+2b^2 &\geq0\\ 2a^2-7ab+2b^2 &\geq -3ab \end{align}

or

\begin{align} 2a^2-7ab+2b^2 &\geq -3ab \\ 2a^2-4ab+2b^2 &\geq0\\ a^2-2ab+b^2 &\geq0\\ (a-b)^2& \geq0. \end{align}

Instead, you'll see English. That's because the point of a proof is to communicate a reason why the theorem is correct. Most often, the best way to communicate is using words, not only formulas. In particular, what you have written is just a list of equations, and if the equations were more complicated, the reader might not even realize that they're supposed to imply each other, one way or another. Of course writing an English sentence between every equation is overdoing it, but in almost any mathematical proof, the author explains in words the idea behind what they're doing.

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With that in mind, I'll answer your first question: You got 0 marks because your proof failed to communicate. As others have said, usually it's common that a series of equations without any explanation means that the first equation implies the next one and so on. Your proof doesn't follow that tradition, which may be ok in many cases, and you didn't explain in any other way what you're doing.

I argued however that you can go back and forth between my operations (if and only if, which doesn't work for the −5=5 example). He still didn't give me the marks for it.

The crucial part of the communication in the proof happened here. Outside your written answer, after it had been graded. Most professors, I think, grade only the answers in the form they're handed in. Of course you're allowed to argue a wrong grading and explain if the professor misunderstands something, but that wasn't the case here: You weren't explaining your existing answer, you added something crucial to your answer.

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(I'd personally have given you somewhere between 0 and 1 marks for having most elements of a correct answer there. But that's always quite subjective, so I can't say your professor was wrong to give you 0, I don't know if your they give half marks.)

Answer 6

The following is touched on in some answers and comments, but I feel is worthy of a separate answer.

The primary purpose of assignments and exercises like this is not (simply) to get you to produce the correct answer but to demonstrate to your instructor that you both understand the concepts involved and can convey your understanding of those concepts.

Starting with a false statement you can "prove" anything. Starting with a potentially false statement (or, at least, one whose truth is unknown) will lead to an erroneous proof unless you are able to say (and do say) that each step (ultimately leading to a "known true" statement) is an example of if-and-only-if or implies-and-is-implied-by ($\iff$).

Thus, in your answer, even though – as you say – "Dividing by 2, adding and factoring are pretty obvious right?" because you have not explicitly stated that each step is reversible, your instructor does not know whether:

1. You are aware of the dangers of a "proof from fallacy" but have not bothered to make this clear, or:

2. You are blissfully unaware of those dangers, and – while you got lucky this time – you might use $-5=5 \Rightarrow 25=25$ next time and not notice the proof is invalid.

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In a similar vein, when I was at school we were always told to show our full working, because only then could the teachers determine whether we understood what were doing or not. There were several outcomes:

• Incorrect answer; no working: Zero marks.

• Correct answer; no working: Limited number of marks (probably no more than half the total).

• Incorrect answer; workings shown: If the workings demonstrated the correct method, but included a simple mistake, you were likely to get reasonable marks (possibly three-quarters).

• Correct answer; workings shown: Assuming the correct method was used, full marks.

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To answer your specific questions:

1. Should I have received marks for this question?
   _{(original question; as originally answered)}

In this case, I'd be tempted to agree with your instructor. Having chosen a "non-standard" layout, and without explicitly indicating you are aware of the dangers of a "proof from fallacy", while a little harsh, I don't think zero is overly harsh. The fact that it has stuck with you sufficiently to come here and ask about it is evidence that you won't make the same mistake again which is really the main goal of assignments like this.

1. Is my proof equally valid?
   _{(after question edited by other than OP, although I believe the original question is more relevant)}

Other answers directly tackle the "correctness" of the OP's proof and I don't really have anything to add (other than a personal view that the assumed if-and-only-if nature of each step should have been made more explicit).

The point of my answer is that this question is not what someone in the OP's position should be asking (and, to be fair, the OP didn't actually ask this question). What is important is not whether the proof is correct, but whether the student has demonstrated their understanding of the topic.

So: the proof may be correct, but it's still a bad answer to the exercise.

2. Do real mathematicians all write one way or the other when writing in a paper?

I am not familiar enough with the writing of academic mathematical papers to be certain, but I would expect (hope?) that were someone to present a proof in the form you did, they would be explicit in their use of $\iff$ between stages (or, more probably, would have used the "conventional" layout).

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_{(An addendum after responding to the edited question.)}

The premise of my answer is that tests/exercises like this are predominately about a student demonstrating that they understand the topic at hand. When I was at school in the UK, studying for what then were O Levels, this was drummed into us constantly: getting the right answer is less important than showing how you got it.

You may possibly have a legitimate cause for complaint if your instructors have not sufficiently emphasised this point in the past.

If you feel that this is the case, I'd recommend a conciliatory (rather than confrontational) approach. Explain to your instructor that you now understand why they felt unable to give you marks for this question – i.e. because your approach leaves room for not understanding the problems of proof by fallacy — but that you feel this need to demonstrate such understanding hadn't been sufficiently "driven home" in the past, and could they (the instructor) focus more on this aspect in the future.

Answer 7

The following proof would work. So you can write it backwards but you need the arrows to show that you are going backwards.

\begin{align}2a^2-7ab+2b^2 \geq -3ab& \impliedby 2a^2-4ab+2b^2\geq0\\&\impliedby a^2-2ab+b^2\ge 0\\&\impliedby(a-b)^2\ge0\end{align}

Learning to get connectives right is an essential part of your study so don't be surprised if you lose marks for getting them wrong.

Answer 8

As others have said, your argument is ambiguous enough that it probably didn't deserve full marks. But I'd hardly call it incorrect. In future, the phrase "the following are equivalent" can be used to resolve these kinds of ambiguities.

Consider $a,b \in \mathbb{R}$. Then the following are equivalent:

$2a^2-4ab+2b^2\geq0$

$a^2-2ab+b^2\geq0$

$(a-b)^2\geq0$

$\mathrm{true}$

Hence $2a^2-4ab+2b^2 \geq 0$ for all $a,b \in \mathbb{R}$.

A useful shorthand is TFAE.

Consider $a,b \in \mathbb{R}$. Then TFAE:

(whatever)

Another approach is to put the word "show" in front of any formula you're not assuming, but rather, trying to prove. For example:

Consider $a,b \in \mathbb{R}$.

We're required to show $2a^2-4ab+2b^2\geq 0.$

So it suffices to show $a^2-2ab+b^2\geq 0$

So it suffices $(a-b)^2\geq 0$

But this is true.

Hence $2a^2-4ab+2b^2\geq0$ for all $a,b \in \mathbb{R}$.

A useful shorthand us just using the word "show" without all the English fluff. For example:

Consider $a,b \in \mathbb{R}$.

Show $2a^2-4ab+2b^2\geq 0.$

Show $a^2-2ab+b^2\geq 0$

Show $(a-b)^2\geq 0$

Show $\mathrm{true}$

Hence $2a^2-4ab+2b^2\geq0$ for all $a,b \in \mathbb{R}$.

Answer 9

I was taught to proove inequalities using the transitivity property of inequality relation.

For example: $$2a^2-7ab+2b^2 \geq 2a^2-4ab+2b^2 - 3ab \geq 2(a-b)^2 - 3ab \geq 0-3ab \geq -3ab$$

Answer 10

I would say that your kind of backtracking is a good approach that will give you a sketch of a proof. After the sketching you must construct a logical correct proof, in this case just prove all steps the other way, from
$(a-b)^2\ge 0\quad$ to $\quad2a^2−7ab+2b^2≥−3ab$.

Answer 11

The teacher is correct, but hasn't explained it well.

Working one way doesn't always prove the opposite way. This is less obvious in basic maths, but it's important to grasp. Why it doesn't becomes more obvious when you consider the kinds of operations used. With operations like inequalities, complex numbers, set inclusion and undefined operations like division (by zero), it is often the case that what works one way just doesn't prove anything about what works the other way, unless you actually check it will work the other way. But you have to specifically check that.

Whichever way the answer comes to mind, you will have to work it out the right way to actually prove anything (rather than just suspect it)