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"Everybody knows" that $a\ll b$ means a quite vague thing, something like $a$ is very much less than $b$.

(And on math.stackexchange.com, it may be observed that not everybody knows the difference in MathJax and LaTeX code between $a\ll b$ and $a<<b$.)

My question is whether there is a conventional notation for a similar concept, but which I will define precisely below, and if there's not, then what would be a good notation for it? For now I'll use the notation $x\preceq a$.

Its precise definition is this: $$ P\text{ holds for }x\preceq a \tag 1 $$ $$ \text{means} $$ $$ \text{for some }\varepsilon>0,\text{ for all }x\in(a-\varepsilon,a),\text{ $P$ holds.} $$ One could express this as saying $P$ holds for $x$ not much less than $a$, and this gives a precise definition to that concept. But notice that $x$ and $a$ do not play symmetrical roles but with the direction of the inequality reversed, i.e. this does not mean the same thing as "$P$ holds for $a$ not much more than $x$". Probably it would be a good idea to have a reminder of that asymmetry in the notation. And of course I'd like to keep it simple.

So we want something

  • as short and simple as line $(1)$ above (in particular, temporarily sweeping under the carpet all attention to the quantity $\varepsilon$ and the quantifiers $\forall$ and $\exists$), but
  • with the needed suggestive asymmetry, and
  • not too easlily confusable with other frequently seen conventional notations that have different meanings.
Michael Hardy
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3 Answers3

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Is it more the less-than aspect that is key, or the nearness? I wonder if

$$x \mathbin{{\approx}_{-}} a$$

might work, by analogy to the $\lim_{x\to a^{-}}$ operator, though this highlights the nearness more than the ordering.

Michael Hardy
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wonder
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We could write $$ P \text{ holds } \forall x \in {}_{\varepsilon}{a} $$ to mean

for some $\varepsilon>0$, for all $x\in(a-\varepsilon,a)$, $P$ holds.

Of course it makes sense to define the same notion on the right: write $$ P \text{ holds } \forall x \in a_\varepsilon $$ to mean

for some $\varepsilon>0$, for all $x\in(a,a+\varepsilon)$, $P$ holds.

Abramo
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  • I'm not sure this is worth doing unless it's really a lot simpler than writing out "for some $\varepsilon>0$, for all $x\in(a-\varepsilon,a),\text{ $P$ holds}$." Your notation still seems too complicated. – Michael Hardy Dec 08 '13 at 20:25
  • Maybe I didn't get your point, but to write "$P$ holds $\forall x\in {}_\varepsilon a$" is much shorter than "for some $\varepsilon>0$, for all $x\in(a−\varepsilon,a)$, $P$ holds." – Abramo Dec 08 '13 at 20:51
  • Shorter in actual length, yes. Maybe what I want is a notation that temporarily sweeps under the carpet all attention to $\varepsilon$. – Michael Hardy Dec 08 '13 at 21:00
  • Ok, then I didn't get it. Maybe you should write this requirement explicitly in your question ;-) – Abramo Dec 08 '13 at 21:01
  • ...besides, the letter $\varepsilon$ would normally mean some number, making the meaning of "$\forall x\in a_\varepsilon$" dependent on the value of $\varepsilon$. That of course is not what you mean by it, but that makes me uncomfortable. – Michael Hardy Dec 08 '13 at 21:02
  • I thought my "short and simple" desideratum did that. That there were these particular complications in the definition of simplicity had not yet occurred to me. – Michael Hardy Dec 08 '13 at 21:03
  • Maybe you could use something like "$P$ holds in ${}_\bullet a$" – Abramo Dec 08 '13 at 21:03
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    Now you're bringing to my attention another desideratum: I'd like the reader to be temporarily distracted from the whole issue of the precise meaning so that while reading "$P$ holds for $x\preceq a$", the attention would be more on the meaning of $P$ (which might in itself be fairly involved) and on $x$ being a little bit less than $a$ than on just what that last thing means. If the reader sees ${}_\bullet a$, the reader will be thinking "I remember: he said that means thus-and-so". A sort of mental speed bump---just what I want the notation to avoid. – Michael Hardy Dec 08 '13 at 21:08
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How about "$P$ holds in some open left-neighborhood of $a$"?

I think this even agrees with the definitions used in some calculus courses.

Yoni Rozenshein
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  • Well, again, that puts a lot of words into the point I'd like to temporarily distract the reader from. I have in mind an audience of non-mathematicians who are perfectly fluent with mathematical calculations and elementary algebra, such as happen in (for example) finding $\int\sec^3 x,dx$, but less so with logical reasoning such as happens in proofs. – Michael Hardy Dec 08 '13 at 21:12
  • $P$ holds for all $x \in N^-(a)$? (Obviously, $N^-(a)$ is not really a set since it depends on a hidden $\epsilon$ which is implied to exist, so it might be weird notation in some ways, but maybe convenient in others.) – Yoni Rozenshein Dec 08 '13 at 21:17
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    Another possible option is "$P$ holds for $x \to a^-$", which is a notation you can write informal calculus in. For example "$x^2 \approx 4$ for $x \to 2^-$". – Yoni Rozenshein Dec 08 '13 at 21:21