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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?

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    $\begingroup$ General advice: do as your teacher requires. That said, the idea of your proof is right, but you don't make it obvious enough that all steps are equivalences. You would have had a stronger case if each of the ensuing lines were started with an $\iff\,$ (and that's in fact how I would write it, since it flows more naturally). $\endgroup$ – dxiv Sep 17 '17 at 3:26
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    $\begingroup$ Dividing by 2, adding and factoring are pretty obvious right? $\endgroup$ – mtheorylord Sep 17 '17 at 3:27
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    $\begingroup$ It's a good thing that they are obvious to you. But the teacher can't guess (or assume) that unless you spell it out. Many mistakes happen precisely because something looks to be too obvious. $\endgroup$ – dxiv Sep 17 '17 at 3:29
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    $\begingroup$ Interesting that you consider your argument forward. I would say that your argument is backward, as it goes from (what should be) the end of the argument (namely the conclusion you wish to reach), to (what should be) the beginning of the argument (namely something that is known to be true). $\endgroup$ – paw88789 Sep 17 '17 at 9:46
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    $\begingroup$ There is not often enough emphasis on false proofs. If you had proved $1\le0$ by doing the step of multiplying both sides by $0$, of course this wouldn't be valid. To make sure there is no question about dividing by 0 or choosing the wrong square root, it's best to check by writing things in what you call the reverse order. $\endgroup$ – Mark S. Sep 17 '17 at 15:17

11 Answers 11

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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).

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    $\begingroup$ @StephenG I would respectfully disagree. The standards and expectations of mathematics at degree level are very significantly greater than at school level. Mathematics requires exceptional precision and rigour - it's extremely easy to make a tiny, barely visible, innocuous mis-step in a proof and derive the wrong result. This can never be allowed to happen so only the utmost rigour and clarity is accepted in mathematics. OP's mistake is both a common misunderstanding and lacks clarity. It is standard practice and absolutely correct to award zero marks. That's just how mathematics is. $\endgroup$ – niemiro Sep 17 '17 at 13:38
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    $\begingroup$ Mathematical proof is inherently required to be rigourous. Pedantic is just a perjorative way of saying the same thing. The thing about using $\iff$ is that the reader is forced to consider both $\implies$ and $\Longleftarrow$. In this case, that is very little extra effort, but the problem is still there. When not implicitly stated, the flow of a mathematical arguement is $\implies$, which makes the OPs argument incorrect. Reversing that arguement would have been correct and it would not have required the reader to also consider $\Longleftarrow$. $\endgroup$ – steven gregory Sep 17 '17 at 14:15
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    $\begingroup$ @niemiro Worth emphasis: failure to (explicitly) derive bidirectional inferences, and confusion between necessary and sufficient conditions are widespread errors made by beginners. Arguments like those in the OP do not contain enough information to rule out such errors (we can't read the author's mind, nor should we have to). $\endgroup$ – Bill Dubuque Sep 17 '17 at 15:19
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    $\begingroup$ The purpose of the exercise is not just to see whether or not the student prove this inequality correctly. Any exercise which begins "Prove that..." and which is intended to be marked is, implicitly or otherwise, a test of the students ability to communicate effectively. That's the whole point of writing down a proof: to communicate the logic of your argument to another person (or perhaps to your future self). From what the OP has told us, his proof did not communicate the logic of his argument effectively, and therefore he does not deserve the marks in this case. $\endgroup$ – Will R Sep 17 '17 at 16:54
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    $\begingroup$ @Ovi: The thing is that you're failing to distinguish between a proof and the work you do to help you find a proof. The thing you find enlightening is not a proof at all -- it's the work that went into discovering a proof. $\endgroup$ – Hurkyl Sep 18 '17 at 10:47
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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.)

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    $\begingroup$ The first time this really mattered to me was in making a $\delta - \epsilon$ proof of the existance of a limit. I always started by assuming the conclusion and trying to work back to the premise. Having a better idea of what had to be done, I would then figure out how to make things work in the right direction. The "forward" proof was seldom the reversal of the "backward" proof. $\endgroup$ – steven gregory Sep 17 '17 at 14:37
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    $\begingroup$ Without an implication (in the form of "$\implies$" or in words) between the lines, the professor's variant is not really better. Additionally, any implication used should be justified (such as "By expanding the parentheses, ..." or "Multiplying both sides by42$, we get ..."). Both these points should be learned at this early stage. $\endgroup$ – Hagen von Eitzen Sep 18 '17 at 6:42
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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.

  1. 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.

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    $\begingroup$ What you wrote is ... No, what the OP literally wrote is not one-way implications. Per the (currently) accepted answer "there are two ways to interpret ...". It is indeed ambiguous as written, since the OP did not explicitly mark the steps as two-way equivalences (which they are), and that's most likely why the professor rightfully objected. But OP's fault is one of redacting the (otherwise fine) proof, rather than some deeper flaw about forward vs. backwards collation of the steps. $\endgroup$ – dxiv Sep 17 '17 at 7:22
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    $\begingroup$ @dxiv: fair point - and fixed. I imagine that not assuming equivalence marks got hammered into me at some point in my education, to the point where I hardly even consider them unless explicit now. $\endgroup$ – Denis de Bernardy Sep 17 '17 at 7:31
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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.

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  • $\begingroup$ True, but the art of building fairy castles is common in mathematics. Proof by contradiction is, in essence, making the reader be "in the dark" about the truth of all statements until you show that their truth leads to a contradiction. The point that it is about persuasive communication is good, and yes order can help; but core problem isn't the lack of order but rather the convention that one statement after another is a "implies" relationship and not "logically equivalent to". $\endgroup$ – Yakk Sep 18 '17 at 13:34
  • $\begingroup$ @Yakk: I concur that it's a stylistic technique, and there are other logically correct options. But most people agree that direct proofs are easier to read than proofs by contradiction, so I would argue that we should try to work with that when feasible. $\endgroup$ – Daniel R. Collins Sep 18 '17 at 22:30
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    $\begingroup$ @Yakk To be fair, (good) proof by contradictions tell you right off the bat when they make an assertion they know is false. They might keep you in the dark as to why, but you shouldn't have to wait until the last line to realize assumption A was false the whole time. I think the point is that OP's proof opens with an ambiguous statement and works it way to something true, but it's much easier to define right away what's true and what isn't. Otherwise you kinda have to take the author's word until you reach a statement you're sure of. $\endgroup$ – Lord Farquaad Sep 19 '17 at 21:23
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I'll start by answering your second question:

  1. 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.


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.


(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.)

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  • $\begingroup$ This is the better answer. There is no uniform way to write these proofs and ultimately the professor's decision was an arbitrary one; there is nothing wrong with reducing to a known case rather than building up from a known case, except that the set of forbidden operations is inverted (you cannot multiply by negative numbers or $0$, rather than you cannot multiply by negative numbers or $\infty$). But if your preferences contradict your professor's preferences then there is friction in communication. $\endgroup$ – CR Drost Sep 19 '17 at 14:52
  • $\begingroup$ Like, had one just written $A<B$ instead of $A\ge B$ throughout, one would have a perfectly valid argument if one ends with the sentence "This is a contradiction, therefore $A\ge B$" at the end. It's all about how you frame it, and the problem was that you didn't frame it in the first place. $\endgroup$ – CR Drost Sep 19 '17 at 14:55
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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.


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.


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.

  1. 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).


(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.

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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.

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

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  • $\begingroup$ I would +1 this if it weren't for the abbreviated versions, which I find very hard to parse (and not anything I've commonly seen). $\endgroup$ – Daniel R. Collins Sep 22 '17 at 21:43
  • $\begingroup$ @DanielR.Collins, the abbreviated versions are very important. For example, if you don't offer abbreviated versions, high school students end up not adopting what you're offering, because they can't be bothered to write out the full English sentences. Good mathematics teaching, I think, depends on being able to offer useful abbreviations. That's my opinion. $\endgroup$ – goblin Sep 23 '17 at 8:51
  • $\begingroup$ But there's an enormous difference in meaning between "show" (a command with implied subject "you", the reader) and "it suffices to show". Arguably one is further damaging students' English to act like those mean the same thing. $\endgroup$ – Daniel R. Collins Sep 23 '17 at 15:24
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I was taught to proove inequalities using the transitivity property of inequality relation.

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

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  • $\begingroup$ I think this is really the way on should do it. $\endgroup$ – Glostas Sep 18 '17 at 14:17
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    $\begingroup$ -1 This doesn't answer the question (or either of the questions). $\endgroup$ – JiK Sep 19 '17 at 8:54
  • $\begingroup$ Hm, that step 2(a-b)^2 = 0 is not intuitive at all... $\endgroup$ – Betlista Sep 19 '17 at 12:25
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    $\begingroup$ @Betlista - this is not a chain of equalities, but chain of inequalities. $\endgroup$ – Jaroslaw Matlak Sep 19 '17 at 13:44
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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 $\quad2a^2−7ab+2b^2≥−3ab$.

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  • $\begingroup$ We only have the students version of the professors argument. $\endgroup$ – Philip Roe Sep 19 '17 at 20:08
  • $\begingroup$ @PhilipRoe, yes but from what we know the professors argument was formally correct but maybe not enough constructive. $\endgroup$ – Lehs Sep 20 '17 at 10:50
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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)

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