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From reading some instructions I could find online, I could understand that this isn't universally agreed upon, but in a course I'm taking now the professor insists on a particular connection between the proposition one needs to prove and stating the hypothesis (the opening statement of the proof). However, the class I'm taking isn't in English, and I'm struggling to properly translate it.

So, for example, my professor says that if the proposition includes a "for all x", then the hypothesis should begin with something that I'd interpret as "let x". And, if the proposition says "there exists" and the hypothesis should say something like "look at candidate", and I'm not sure how to translate the later.

wvxvw
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  • I don't understand the question. Can you give concrete examples of the statement of the theorem, and the candidates you are considering for how to begin the proof? – user7530 Feb 13 '15 at 18:54
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    "Look at candidate" sounds like it would normally be "Consider": eg. "Consider $x$ such that $x \equiv q \bmod r$..." etc. – Joffan Feb 13 '15 at 18:55
  • @user7530 example: Prove or disprove: There exists set $A$ such that $P(A)=\emptyset$. Then the way the prof. suggests to begin the proof would be something like "Let A, look at $\emptyset$, since empty set is a subset of any set ... – wvxvw Feb 13 '15 at 18:59
  • I use "let" in the case you described, where the result you are trying to prove makes a "for all" claim. I also use "let" and "define" somewhat interchangeably if I need to define a new variable or other object during a proof. – Ducky Feb 13 '15 at 19:05
  • @Joffan oh, that does sound like it! I would still be glad if I could find some definitive guide as to when to use "let" and "put" (even though I know some people may not agree on details, I think of it as being similar to coding style in programming--a way to be consistent in writing one's programs, and, even sometimes without much understanding it helps other to ease into reading one's code). – wvxvw Feb 13 '15 at 19:07
  • As with all questions of writing style, it is very hard to give universal rules that will apply in every case. "Consider," "let," and "put" are more or less interchangeable in the setting you're describing but the different options do carry nuances. – user7530 Feb 13 '15 at 19:08
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    I'd strongly recommend reading The Most Common Errors in Undergraduate Mathematics. It will likely address everything you have in mind and then some. – Daniel W. Farlow Feb 13 '15 at 19:58

1 Answers1

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If I am proposing a particular value for a variable that has already been named in a problem statement, I would use "consider." Especially if I am providing a counterexample do a universal quantified variable.

Prove or disprove: If $a\vert bc$, then $a\vert b$ or $a\vert c$.

Consider $a=4, b=2,c=2$. Then $4\vert 4$, but $4$ does not divide $2$.

Prove or disprove: $f(x) = \frac{d}{dx}\int_0^x f(y)\, dy$ for every integrable function $f$.

Consider $f=\begin{cases}1, &x=0\\0, &x\neq 0.\end{cases}$ We have that...

"Consider" can also be used when you supplying a constructed value for an existence proof:

Theorem: There exists an integer $n$ with $n^2-4=0$.

Proof: Consider $n=2$. By direct computation we have $2^2-4 = 4-4=0,$ as desired.

"Put" or "take" can also be used in this case:

Proof: Put $n=2$. Then by direct computation we have $2^2-4 = 4-4=0,$ as desired.

But neither of these are completely idiomatic, to my ear. I would use a more verbose construction, e.g.

Proof: We verify by direct computation that $n=2$ is a solution: $2^2-4=4-4=0.$

Finally, I would use "let" when I am naming a new quantity that didn't appear in the theorem statement.

Theorem: Every integer greater than 1 has a prime divisor.

Proof: Let $S$ be the set of integers greater than one that do not have a prime divisor. Suppose for contradiction that $S$ is nonempty; then let $d$ be the least element of $S$...

user7530
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