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In most of mathematics problems you see "prove that..." or "show that..." statement. My question is what's the difference between these two words? I mean is "showing" something different from "proving" it in mathematics?

Thanks.

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There's no difference as far as I know. –  Qiaochu Yuan May 30 '11 at 10:52
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"Show" sounds less intimidating. Curiously, 19th century English problems often used "shew." –  André Nicolas May 30 '11 at 11:00
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There is no difference between "show" and "prove". However, some people think "prove" is very fancy. For example, once when I was a student, our teacher instructed us to do some exercises in a textbook. Exercises 1-9 were of the form "compute the powers of the matrix $A=\cdots$" and exercise 10 was "prove that the powers of the matrix $B=\cdots$ have the form $\dots$". Our teacher said: "exercise 10 is a challenging one because it covers proofs ... this concept usually isn't in the curriculum ... but it's especially important to do this one ... it tells us who the good students are." –  Amitesh Datta May 30 '11 at 12:49
    
Side comment: surprisingly many students use "proof" as a verb, as in "I will proof ....". There are ordinary language precedents ("proof bread dough"). –  André Nicolas May 30 '11 at 13:46
    
@user6312: But that's a different meaning of the word: "to test; examine for flaws, errors, etc.; check against a standard or standards". –  Jefromi May 30 '11 at 15:09
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7 Answers

up vote 16 down vote accepted

Sometimes students misinterpret show to mean give an example. I now avoid using show in exams; I always use prove when a proof is required.

In the context of examples or calculations, it might be ok use show. For instance, "Show that $2$ is a root of $x^2-4$" or "Show that $\sin x$ is a solution of $y''= -y$.

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Along the lines of your second point, I would also use "show" to mean "exhibit an example proving that [some counterexample exists or existence theorem holds]." –  user7530 Apr 23 '13 at 21:38
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No difference. Just a way to fool students...

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Exactly my point. –  lhf May 30 '11 at 12:09
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+1 for showing ( but not proving :) lhf's point! –  Arjang May 30 '11 at 12:25
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No difference. Just another way of stating a question.

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One of my colleagues related his experience. He used "show" on his exams in a certain course thinking it a less intimidating word than "prove". But, as usual, before the next exam the students asked: "Are there any of those problems where you have to show something?" –  GEdgar Jun 2 '11 at 13:17
    
@GEdgar: :) ${}$ –  Mariano Suárez-Alvarez Jul 4 '11 at 17:42
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In Engelking's book "general topology" he uses "show" for easy proofs/examples, and "prove" for the hard ones. He also uses "note" (in exercises) for basically two line observations, often just for later reference (as exercises are an integral part of the text). He does explain this in advance, by no means this is standard. I do think it's convenient. It's a bit like Knuth numerically rating exercises in AoP.

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As noted above, they can be used interchangeable with "show" generally being used for more straightforward proofs while "proof" for more abstract or difficult ones. They are often times used to separate existential versus universal conditions. That is an author is more likely to say "show there is a bounded real-valued function on $\mathbb{R}$ that is not Lebesgue measurable" versus "prove that for all quadratic polynomials $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$". Again, the words "show" and "prove" are interchangeable ("prove there is a ..." or "show that for all..."), but they are commonly used in that way. Or at least one or more of the author's I'm using now uses them that way.

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I remembered when I was in high school my maths teacher said specifically there was a difference between showing and proving.

He said at our - high school - level, everything is generally considered a 'show' which are typically quite specific and do not involve abstract theories; whereas once you hit the higher levels, then you'd start with proofs which are considerably more rigorous. e.g. prove that 1+1 = 2.

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To show is to use numbers to demonstrate a certain property, to prove it to use letters to prove that the system works for all number systems. There is a clear difference that can really hurt you in exam questions and assignments. Trust me, I know.

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