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Others have expressed confusion about this proof on Stack and I have looked through every one of them because I don't want to post a duplicate post, however, none of them answer the questions I have and I am still confused.

So we have to find what is wrong with this proof:

We introduce the following notation, for positive integers $x$ and $y$:

$$\text{max}(x, y) = {x \space \text{if}\space x \ge y, y \space \text{if}\space x < y}$$

What is wrong with the following “proof by induction”?

“Theorem:” For every positive integer $n$, if $x$ and $y$ are positive integers with $max(x, y)$ = $n$,then $x = y$.

Basis step: If $max(x, y) = 1$ and $x$ and $y$ are positive integers, then we have $x = 1$ and $y = 1$.

Inductive step:

Let $k$ be a positive integer, and assume that whenever $max(x, y) = k$ and $x$ and $y$ are positive integers, then $x = y$ (this is the IH). Now let $\text{max} (x, y) = k+1$, where $x$ and $y$ are positive integers. Then $\text{max}(x − 1, y − 1) = k$, so by the IH, we have that $x − 1 = y − 1$. By adding $1$ to both sides we obtain that $x = y$, completing the inductive step.

The answer as to why this proof is wrong is because:

The mistake is in the inductive step: The IH says that whenever $\text{max}(x, y) = k$ and $x$ and $y$ are positive integers, then $x = y$. Now consider the following case: $k + 1 = 2$, $x = 1$ and $y = 2$. Then $x$ and $y$ are positive integers and $\text{max}(x, y) = \text{max}(1, 2) = 2 = k + 1$. Then $\text{max}(x − 1, y − 1) = \text{max}(0, 1) = 1 = k$. But $0$ is NOT a positive integer, so the IH does NOT apply.

Questions I have:

  1. What I don't understand is why does the notation for max$(x,y)$ say that x has to be $\ge y$ and $x$ has to be $< y$ if it explicitly states in the theorem that $x$ has to $= y$. This is a contradiction

  2. In the answer it says to consider the case where $x=1$ and $y=2$ but we CAN'T have that case because $x$ must $= y$, so why did the answer even consider that case

  3. I don't even understand what the theorem means tbh, how does it mean 'all positive integers are equal'. To be it says that for every positive integer $n$, the maximum of $2$ positive integers is equal to $n$. So for example, for the positive number $10, a = 10$ and $b = 10$. For the positive number $11, a = 11$ and $b = 11$. The proof seems correct to me because for every positive integer, the max of 2 numbers which are the same would return $1$ number$\dots n$. So how is this saying that all positive numbers are equal?

I'm just confused by this question in general, can anyone please clear this up for me. Thanks in advance

Blue
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  • The issue is that you can't subtract $1$ from $1$ without getting a non-positive value. Your claim ONLY works for positive integers, so if we have $x=1,y=2$, then $\max(0,1)=1$, but this doesn't imply that $0=1$ since $0$ is not positive. – Rushabh Mehta Aug 25 '21 at 18:31
  • I get the explanation but that isn't where my confusion lies, it's with the 3 questions above @DonThousand – computerscienceisapain Aug 25 '21 at 18:32
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    The theorem is false and the proof is incorrect for the reasons already shown. The purpose of the problem was to showcase an incorrect statement and a *seemingly correct* proof of the obviously incorrect statement so as to allow you to inspect the proof more closely and find where the mistake was. The obviously incorrect statement was that every positive integer is equal to every other positive integer... which would imply that $1$ is equal to $2$ for example... but we clearly know that this is false and so there must have been a mistake in the proof. – JMoravitz Aug 25 '21 at 18:34
  • say hypothetically speaking 0 was allowed, clearly 0 and 1 aren't equal so the theorem is false @DonThousand – computerscienceisapain Aug 25 '21 at 18:36
  • For completions sake, here is a link to another question about this problem: [All natural numbers are equal?](https://math.stackexchange.com/questions/342321/all-natural-numbers-are-equal?rq=1) – JMoravitz Aug 25 '21 at 18:45
  • Also, to emphasize point (2)... that $x$ should equal $y$ is the *conclusion* that the proof reaches... not the hypothesis. That we could try to use the theorem in the case of $x=1$ and $y=2$ and witness the fact that $1$ is in fact *not* equal to $2$ despite our supposed proof saying that despite this we should have come to the conclusion that they *were* equal is all the evidence we need to confirm to ourselves that the proof has an error in it. – JMoravitz Aug 25 '21 at 18:47
  • but x is not bigger than y (the notation states that x must be > y) so how can we use x=1 and y=2? @JMoravitz – computerscienceisapain Aug 25 '21 at 19:03
  • @computerscienceisapain Then your base case fails. You can't have it both ways. – Rushabh Mehta Aug 25 '21 at 19:06
  • no because I am still confused by my third question in my OP which no one has answered yet @MartinR – computerscienceisapain Aug 25 '21 at 19:08
  • @computerscienceisapain Your third question is incomprehensible, as written. – Rushabh Mehta Aug 25 '21 at 19:11
  • what I mean is how does the theorem translate to 'all positive numbers are equal' in english? Simply. I don't understand why this is incomprehensible? @DonThousand – computerscienceisapain Aug 25 '21 at 19:13
  • in the future I will try and word my questions better and make them more specific and focus on one thing to avoid long discussions @amWhy – computerscienceisapain Aug 25 '21 at 19:28
  • Good; I deleted my comment above; It is good you kept trying to pin down what you needed to understand. No problem! – amWhy Aug 25 '21 at 19:32
  • @computerscienceisapain Just in case, more discussion can also be made in chat, if needed. And then editing the question or answers accordingly once things have been clarified. – user Aug 25 '21 at 19:48
  • @MartinR As you can see form the discussion, the given link is only a partial duplicate. The asker here has raised some more specific question. For this reason it shouldn't be closed. – user Aug 25 '21 at 19:51
  • "*but x is not bigger than y (the notation states that x must be > y) so how can we use x=1 and y=2?*" Where do you think you see that? In the definition of max? Do you not know what max is or means? Which number is bigger? 3 or 6? You should know the answer is 6. Which number is bigger 10 or 2? You should know the answer is 10. Which number is bigger 5 or 5? The answer here is 5... not that it the first is strictly bigger than the second... but being tied is good enough. This question of "which is bigger" is what the $\max$ function is... and it can be written and defined as above. – JMoravitz Aug 25 '21 at 23:47
  • "*but x is not bigger than y... so how can we use x=1 and y=2?*" We don't need $x$ to be bigger than $y$... we can ask the question "Which is bigger... 1 or 2?" just as well with 1 and 2 as we could have asked the question with any other pair of numbers – JMoravitz Aug 25 '21 at 23:48

3 Answers3

2

Your third question is

I don't even understand what the theorem means tbh, how does it mean 'all positive integers are equal'.

The false theorem is

“Theorem:” For every positive integer n, if x and y are positive integers with $\,\max(x,y) = n,\,$ then $\,x=y.$

Given any positive integers $\,x\,$ and $\,y,\,$ by the definition of $\,\max\,$ we get that $\,\max(x,y)=n\,$ for some positive integer $\,n.\,$ Then the theorem applies and implies that $\,x=y\,$ which states that $\,x\,$ is equal to $\,y.\,$ But this applies to all integers $\,x\,$ and $\,y\,$ as given to us. Thus, 'all positive integers are equal'.

Somos
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Tha fallacy is in the induction step, when we consider $\max(x − 1, y − 1) = k$ we need that $x,y >1$ but the base case was proved for $x=y=1$.

Then, to proceed by induction, we should verify the base case for $n=2$, which can't be true.

Se also the related:


For your more specific question:

  1. The theorem is indeed false
  2. This is indeed where the base case fails
  3. The problem is constructed considering a false theorem proved by an apparently correct induction. This is a standard exercise to better understand as induction works.
user
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  • I know this is a silly question but I am trying to understand what the theorem is actually stating, what does it actually mean? How does it mean 'all positive numbers are equal'? – computerscienceisapain Aug 25 '21 at 19:05
  • @computerscienceisapain The theorem states something clearly false (as for the example I've linked about horses). The scope is to use induction (in a not correct way) to prove that the theorem is true. This is an exercise to better understand as induction works. – user Aug 25 '21 at 19:11
  • yes but how does it translate to all positive numbers are equal? simple question – computerscienceisapain Aug 25 '21 at 19:14
  • @computerscienceisapain The thesis of the theorem, for the given hypoteses, is that $x=y$ for any $x,y>0$ integers. – user Aug 25 '21 at 19:16
  • okay I agree with that but why is using the Max function relevant to the thesis of the theorem? – computerscienceisapain Aug 25 '21 at 19:17
  • @computerscienceisapain This fact is used to construct the false proof by induction. – user Aug 25 '21 at 19:18
  • @computerscienceisapain please don't expect answerers to spend more and more time to clear things up for you. You're asking beyond what your question asked. – amWhy Aug 25 '21 at 19:19
  • Nice job, @user. – amWhy Aug 25 '21 at 19:20
  • @amWhy In fact the doubt by the asker is very particular in this case. But it always interesting try to understand the different ways to intepret questions form different users. I hope the effort can help in some way! Cheers – user Aug 25 '21 at 19:21
  • Good attitude, @user. Cheers! – amWhy Aug 25 '21 at 19:23
  • apologies @amWhy but I was just letting them know that my main confusion was in my third question addressed in the OP so it was expanding on the third question. But thank you for the help anyway user I appreciate it :) – computerscienceisapain Aug 25 '21 at 19:26
  • I understand, @computerscienceisapain. I get it. You did fine. No problems. – amWhy Aug 25 '21 at 19:30
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The induction hypothesis is only applicable for positive integers $x$ and $y$. However, it is not guaranteed that $x-1$ and $y-1$ are both positive (this is where the counter-example comes into play).

Moreover, the counter-example is valid, since it satisfies all the conditions required by the "theorem." Sure, the theorem's conclusion ($x=y$) is wrong, but that's exactly why this constitutes a counter-example!

Zuy
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  • could you please address my third question, what is the theorem trying to say? I know it is false but I can't understand the explanations if I don't know what it means? How does it mean 'all positive numbers are equal'? – computerscienceisapain Aug 25 '21 at 19:01
  • Nice job answering, @Zuy. – amWhy Aug 25 '21 at 19:21