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I've read related questions but I couldn't find a clear explanation as to, for example, when one should use "...an arbitrary but fixed...". Does it mean that my variable has a certain property so it's not any variable? but most importantly, When should one use it?

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    Could you give an example with some more context? – Stephen Donovan Jul 05 '21 at 02:14
  • When I read the principle of mathematical induction in "Calculus" by Tom Apostol in the induction step reads: "For an arbitrary but fixed $k$, suppose $p(k)$ and show $p(k+1)$ is true. What does it mean and when should one use it? @StephenDonovan –  Jul 05 '21 at 02:22
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    The phrase “arbitrary but fixed” means that it is some global constant, but the constant could be anything. As an example, we could say, “Let $A$ be an arbitrary but fixed constant. Then the derivative of $\sin(Ax)$ is $A\cos(Ax)$.” – Clayton Jul 05 '21 at 02:23
  • To elaborate a little on Clayton's comment, you'd write something like this when you want to prove that a statement holds for all elements of a specific set. For example, you want to prove that for any real number $r,$ the inequality $r^2 + 1\geq 0$ holds, or that for any positive integer $n,$ $1 + 2 + \cdots + n = \frac{n(n+1)}{2}$ is true. Then to do so, you want to consider a general real number $r$ or a general positive integer $n$ and show that the result is true for that general $r$ or $n.$ This would prove that the result always holds, because $r$ or $n$ was completely general. – Stahl Jul 05 '21 at 02:25
  • See also this question. – Stahl Jul 05 '21 at 02:26
  • @Stahl In your example "arbitrary" and "fixed but arbitrary" are the same thing, aren't they? –  Jul 05 '21 at 02:31
  • @Stahl In the linked question I couldn't find an answer. –  Jul 05 '21 at 02:31
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    @Clayton May you tell me when I should used "arbitrary but fixed" instead of just "arbitrary"? –  Jul 05 '21 at 02:32
  • Most of the time, I only use “arbitrary” when I plan on using it in a limit or some other operation that involves the parameter changing. I think this is personal preference, though; mathematically, I think the two phrases are interchangeable. – Clayton Jul 05 '21 at 02:37
  • If they are interchangeable I could, for instance, in an induction proof make my induction hypothesis something like: "suppose for any $k$, $p(k)$ is true"? Because I've seen videos that emphasize $k$ is not arbitrary since we would be assuming what we are trying to prove? –  Jul 05 '21 at 02:43
  • @HannyBoy "arbitrary" and "fixed but arbitrary" probably would generally mean the same thing. This agrees with the interpretation in Qiaochu's answer in the question I linked, and I would agree with what's been said here and there that the two are generally interchangeable. – Stahl Jul 05 '21 at 06:24
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    However, you cannot say in your inductive hypothesis the phrase "suppose for any $k,$ $p(k)$ is true," because this would imply that $p(k)$ is true for all $k,$ which is what you want to prove. However, if your hypothesis is that "fix an arbitrary value $k,$ and suppose that $p(k)$ holds," this is OK, because while $k$ is arbitrary here (meaning it might be any specific choice of $k$), you are only assuming that $p(k)$ is true for one specific value of $k.$ – Stahl Jul 05 '21 at 06:27
  • @Stahl You cleared all of my doubts thus far. You have no idea how grateful I am! :D –  Jul 05 '21 at 17:17
  • @Stahl May you teach me how to link any questions within a comment, please? –  Jul 05 '21 at 17:22
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    I'm glad I could help! To link any url, you can write {your text here}(your url here), and replace the {}'s with []'s. This will display "your text here," and when you click on it, it will take you to the url you put inside the parentheses. – Stahl Jul 06 '21 at 01:18

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Suppose that your job is to prove a statement of the form

For all $x \in S$, $P(x)$

where $P(x)$ is some true-false mathematical sentence.

Here's how you start the proof.

Let $x \in S$. We must prove that $P(x)$ is true...

There are a lot of different ways to reword this in natural language, and one of those ways is

For an arbitrary but fixed $x \in S$, we must prove that $P(x)$ is true...

This has the exact same meaning, as far as the mechanics of proof go.

In your particular example (from the comments) of an induction proof, I would myself word it like this:

Let $k$ be a natural number. We must prove that $p(k)$ implies $p(k+1)$. So, assuming that $p(k)$ is true, we must prove that $p(k+1)$ is also true.

Lee Mosher
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  • So, since I assume $k$ to be arbitrary if say pick up on the fact that $k+1$ is a natural number, could I suppose what we use for $k$ but IN PARTICULAR for $k+1$? I mean since they're interchangeable I wouldn't think otherwise –  Jul 05 '21 at 02:49