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When being asked to justify some conclusion, is it same as to prove it?

My memory tells me that "justify" has been used to describe some informal verification, not necessarily formal proof. I wonder if it is true? If yes, in what sense is "justify" informal? For example, only need to prove necessarity not sufficiency?

Thanks and regards!

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  • $\begingroup$ It probably depends strongly on context. In my experience it means anything from "provide some evidence" to "prove." $\endgroup$ May 30, 2011 at 14:15
  • $\begingroup$ @Qiaochu: Thanks! Do you mean necessary condition of the conclusion by "evidence"? $\endgroup$
    – Tim
    May 30, 2011 at 14:26
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    $\begingroup$ Doesn't that depend on what you're being asked to justify? $\endgroup$ May 30, 2011 at 15:15

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A computation, properly laid out, is of course a proof. However, many students, after years of multiple choice tests, have learned to take the point of view that the answer is the only thing that matters.

"Justify" can be a reminder that the problem will be graded carefully, that (contrary to their usual experience) a slapdash computation will not necessarily get full marks.

I do not think that "justify" carries any connotation of "you need only show necessity but not sufficiency."

"Prove," in a course context, can often mean that a more or less specific set of tools should be used. "Justify" has a more informal feel, but I do not think of it as carrying a lower level of precision.

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  • $\begingroup$ Yes, in hindsight this agrees with my experience as well. I guess one interpretation of "justify" is "make sure to use words." $\endgroup$ May 30, 2011 at 18:16
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To me, "justify" means to lay out the mathematical thought process step by step, so that the line from the starting point to the ending point is connected.

It is a bit less formal than a proof, which has certain logical requirements, but it means, "show enough work so that I know that you get the whole thing."

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"Justify," just like its buddy "show," really means "prove."

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I also encounter "justify", other than as a synonym for "prove", in meta-mathematical discussion. Sometimes (a lot of the time) the author of a book will invent a notation for a particular object being studied. In this case he/she might "justify" the invented notation, which usually means giving a reason why it's not arbitrary.

For example, while the sum of real valued functions is a different operation than the sum of real numbers, the same symbol "+" is used. I don't think this is the best example, but there's a plethora of them if one looks.

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I guess justify means don't assume away.

Some possible exam questions:

1 Prove Borel-Cantelli Lemma.

2 Show that the series given below satisfies the differential equation. Justify all your steps.

3 Show that the series given below satisfies the differential equation. You do not have to justify switching derivative and summation.

In case 2, the professor is saying one cannot assume certain steps are valid as was done in previous classes. In this class, we proved things we assumed away in previous classes. You are to prove them here as well.

In case 3, the professor is saying one can assume said steps are valid.

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