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Is $a≠b≠c$ a shorthand for ($a\neq b$ and $b\neq c$), or is it a shorthand for ($a\neq b$ and $b\neq c$ and $c\neq a$)?

I want to know the "correct and formal" answer, rather than the "most commonly used" answer.

amWhy
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cr001
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    It should not be a short hand for it since $\neq$ is not a transitive relation. That doesn't stop some people from incorrectly using it as a shorthand for this however. The superior way to write it is "$a,b,c$ unique" or some variation. – JMoravitz Aug 13 '20 at 15:20
  • To give a bit more context, the usage comes in this question and is assumed to be the latter case. https://math.stackexchange.com/questions/3787590/prove-that-aw-1-not-perp-bc – cr001 Aug 13 '20 at 15:23
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    At any rate this is a bad notation. There is no "correct and formal" answer, it depends on what convention you follow, and conventions vary. I have once polled a large group (150ish people, mostly math students) about how they would interpret such an expression, and the split was close to even (60% to 40% in favor of "$a\neq b$ and $b\neq c$"). – Wojowu Aug 13 '20 at 15:26
  • I see the issue. The expression itself is informal and is not accepted as a formal writing. Am I correct? – cr001 Aug 13 '20 at 15:28
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    Correct. I don't think there is a fully established convention on chaining nontransitive relations like this. – Wojowu Aug 13 '20 at 15:29
  • Thank you so much. – cr001 Aug 13 '20 at 15:30
  • I would say "no" that is totally unacceptable. If I ever saw $a \ne b \ne c$ I would read it from left to right as "$a \ne b$ and $b\ne c$" (which obviously does not imply $a\ne c$). In fact I'd have to go back and reread it for it occur to me it could be interpreted any other way. But in English that'd be the equivalent of a run-on sentence (ex. "I saw a dog sit next to a cat ate a mouse") – fleablood Aug 13 '20 at 15:36
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    "I don't think there is a fully established convention on chaining nontransitive relations like this" The phrase "$a,b,c$ are pairwise " is pretty standard. Although "pairwise unequal" is a bit awkwardnessful and simply "unique" or "distinct" would be a better choice. – fleablood Aug 13 '20 at 15:38

1 Answers1

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Summarizing comments:

$\neq$ is not a transitive relation. As such, $a\neq b$ and $b\neq c$ does not directly imply that $a\neq c$.

There are some who interpret $a\neq b\neq c$ to imply all of $a,b,c$ are pairwise distinct. There are others who interpret $a\neq b\neq c$ to instead imply only the two statements $a\neq b$ and $b\neq c$ which does not imply anything about the relationship between $a$ and $c$.

In the end, the notation itself is generally considered informal and should be avoided and if you see it used you should infer from context which is meant or ask for clarification.

Suggestions for writing that variables $a_1,a_2,a_3,\dots,a_n$ are all distinct:

  • Order them: "Let $a_1,a_2,\cdots, a_n$ be real numbers such that $a_1<a_2<a_3<\cdots a_n$" (only works in an ordered field)
  • Use words: "Let $a_1,a_2,\cdots, a_n$ be distinct real numbers"
  • Use quantifiers: "Let $a_1,a_2,\cdots, a_n$ be real numbers such that $\forall i\neq j$ we have $a_i\neq a_j$"
  • Write each: "Let $a_1,a_2,a_3$ be real numbers such that $a_1\neq a_2,~a_1\neq a_3$ and $a_2\neq a_3$" (impractical with more variables)
JMoravitz
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