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Proposition: for every positive integer $n$, there do not exist four positive integers $a,b,c,d$ with $ad=bc$ and $n^2 <a<b<c<d<(n+1)^2$

I understand how to prove this by looking for a contradiction when assuming those integers exist. However, how do I translate this into a logic statement without using "$\nexists$"?

Jim Ferry
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Si_monster
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    Hello. Are you aware that "$\nexists x,, p(x)$" is equivalent to "$\forall x,, \neg p(x)$"? (where $\neg$ means "not".) For instance, "There doesn't exist an animal who is a griffin" is equivalent to "Every animal that exists is not-a-griffin". – Stef Aug 17 '23 at 19:00
  • Great, thank you. This is what I initially thought but I wasn't sure. – Si_monster Aug 17 '23 at 19:38

1 Answers1

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Stef provided the key idea in the comments, but let's do it in English. First, for every positive integer $n$, whatever four positive integers $a$, $b$, $c$, and $d$ we select, the following statements cannot both be true: $ad = bc$ and $n^2 < a < b < c < d < (n+1)^2$. This phrasing is a bit awkward, however. Saying "A and B cannot both be true" is the same as saying "if B is true, then A must be false". Using this, we may rewrite the original proposition as follows:

For every positive integer $n$, if the positive integers $a$, $b$, $c$, and $d$ satisfy $n^2 < a < b < c < d < (n+1)^2$, then $ad \ne bc$.

I will leave it to you to translate these ideas expressed in English into ideas expressed in the notation of logic.

Jim Ferry
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