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I have the following statement

$$\neg\exists a\forall b \exists c (ab = b^2c + c)$$

where $a$,$b$,$c$ are all real numbers.

How do I determine whether its true or not, what is a good approach with these type of problems?

Arnold Doveman
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  • I would see if the statement without the negation is true or false ... and just read it to yourself in English: "is there some number $a$ such that for whatever second number $b$ I pick, I can find some third number $c$ such that $ab=b^2c+c$?" And focus on 'interesting' numbers like $0$ and $1$ – Bram28 Oct 09 '17 at 16:13
  • Whoops, missed that it was about real numbers. – Thomas Andrews Oct 09 '17 at 16:15

2 Answers2

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HINT: Standard rules for quantifiers give that your statement is equivalent to: $$ \forall a \exists b \forall c (ab\neq b^2+c) $$ Let us take $a=0$. Is it possible that exists $b$ such that for all $c$ we have $(b^2+1)c\neq0$?

Przemysław Scherwentke
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I assume the negation affects everything inside, so we'll write this as $\neg[\exists a\forall b \exists c (ab = b^2c + c)]$

Let's be a little more tidy with this and write again as $\neg[\forall b, \exists a,c /(ab = b^2c + c)]$

So now we negate everything. For all becomes exists, exists becomes for all, and then you negate the equation. So that's

$$ \forall a,c, \exists b /(ab \neq b^2c + c)$$

Now this is something easier to prove either true or false (I'm betting on false)

Francisco José Letterio
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  • Be careful - it's not generally valid to swap the order of universal and existential quantifiers. – pseudocydonia Oct 09 '17 at 16:51
  • I didn't know that, why is that? In this case I'm swapping all of them ate the same time I negate the equation. In which cases would swapping it be considered wrong? – Francisco José Letterio Oct 09 '17 at 16:53
  • Consider: "for every natural n, there exists a natural m which is greater than n" versus "there exists a natural m, such that for every natural n, m is greater than n." One of these is true about the standard natural numbers and the other is not. – pseudocydonia Oct 09 '17 at 16:57
  • but there is no negation in what you've done. Also those are swapping the order but rather swapping the "such that" part. Let me write what you wrote down below. – Francisco José Letterio Oct 09 '17 at 17:03
  • "for every natural n, there exists a natural m which is greater than n" would be $\forall n \in \mathbb{N} \exists m/m>n$ – Francisco José Letterio Oct 09 '17 at 17:04
  • The other statement is noit switching the order of exists and for all, but rather changing where they are (before/after the "such that"). So your second statement "there exists a natural m, such that for every natural n, m is greater than n." becomes $\exists m \in \mathbb{N}/m>n \forall n \in \mathbb{N}$ – Francisco José Letterio Oct 09 '17 at 17:07
  • The order in words is not the same as the order in logics. I didn't change where the "exists" and "forall" statements are. Your two statements are not the same because when you write them down they're not equivalent. Note that in your first statement the "for all n" comes before the "such that", while in the second one it is after that – Francisco José Letterio Oct 09 '17 at 17:08
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    Sorry, I was being lazy and not using markups. I was trying to say that $\forall n \exists m (m>n)$ is not the same as $\exists m \forall n (m>n)$. – pseudocydonia Oct 09 '17 at 17:09
  • oh sorry, to be honest I never used (or have seen) that kind of notation before. I just thought that the "such that" would come right before the parenthesis because I'm used to that notation. I don't know how the one you and OP are using works – Francisco José Letterio Oct 09 '17 at 17:11
  • In your case, when you say "let's be a bit more tidy" you changed the order of quantification from $\exists a \forall b \exists c$ to $\forall b \exists a,c$, which is strictly weaker - in the first case there needs to be a single $a$ compatible with any choice of $b$. – pseudocydonia Oct 09 '17 at 17:11
  • In particular, compare the final statement you have in your answer with the statement the other poster has given - the order of quantification is different even though the statement on the inside of the quantifiers is the same. – pseudocydonia Oct 09 '17 at 17:13
  • Not a problem - the "such that" is just a filler phrase to make translating logical statements into English sound more natural. – pseudocydonia Oct 09 '17 at 17:13