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I am wondering how to notate "for all positive real value $c$"

Is there a correct notation among the following? $$ \forall c \in \mathbb{R} > 0\\ \forall c \left( \in \mathbb{R} \right) > 0\\ \forall c > 0 \in \mathbb{R}\\ \forall c > 0 \left( \in \mathbb{R} \right)\\ $$

My ultimate goal is notating the following sentence.

"$o(g(n))=\{f(n):$ For any constant positive real value $c$, there is a constant $n_0$ such that $0 \le f(n) \lt cg(n)$ for all $n \ge n_0\}$"

My trial is $$ o(g(n))=\{f(n):\forall c>0(c\in\mathbb R), \exists n_0\in\mathbb{N} \ \ \ \ s.t.\ \forall n>n_0,\ \ 0 \le f(n) \lt cg(n)\} $$

I want to correct this part: $\forall c>0(c\in\mathbb R)$

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    $\begingroup$ Some notations $\mathbb{R}^+_0$, $\left[0,\infty\right)$ $\endgroup$ – GAVD Oct 12 '15 at 8:46
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    $\begingroup$ $\forall c\in \mathbb R,c>0$. $\endgroup$ – Yves Daoust Oct 12 '15 at 8:48
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    $\begingroup$ I would interpret the subscript 0 as indicating that 0 is included in the set, whereas Danny_Kim's question indicates that it isn't. I personally like notation like ${\mathbb R}_{>0}$, which so far as I know isn't standard but I think is clear and unambiguous. $\endgroup$ – Gareth McCaughan Oct 12 '15 at 11:54
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    $\begingroup$ @GarethMcCaughan: I disagree: the notation ${\bf R}_{>0}$ is used by many mathematicians, and clearly understood by all. Which is the only definition of "standard notation" that matters, in my humble opinion. Moreover, it is concise and unambiguous (unlike ${\bf R}_+$ which is rather ambiguous out of context), and at the same time not quite as (formally) nonsensical as the ones proposed by OP. $\endgroup$ – tomasz Oct 12 '15 at 22:27
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    $\begingroup$ As for the question, I think it really depends. When you are actually writing on a white/blackboard, all of these are usable, after some small adjustments (not writing them in one line). When writing an electronic document, all of these are just bad. $\endgroup$ – tomasz Oct 12 '15 at 22:29
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If you would like to have mercy on your reader, please avoid squeezing too many relations together. "$\forall c \in \mathbb{R} > 0$", for example, is readable, but it is not logically precise.

There are at least two ways out; the first one is to say "for all $c \in \mathbb{R}$ such that $c > 0$", and the other is to define the set of all reals $> 0$ and say "for all $c$ in the set ".

You may also use "for all positive $c \in \mathbb{R}$", but this is risky if you do not specify in the first place what your "positive" means; for people may interpret "positive" differently.

In sum, the precise and safe way seems to be "for all $c \in \mathbb{R}$ such that $c > 0$".

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  • $\begingroup$ Thank you for informing to me about precise notation! $\endgroup$ – Danny_Kim Oct 12 '15 at 9:12
  • $\begingroup$ I like everything except this answer except for the "for all $c \in$ the set " bit. The symbol $\in$ really shouldn't be used in place of "in". $\endgroup$ – Git Gud Oct 12 '15 at 9:25
  • $\begingroup$ @Danny_Kim: Thank you; happy to be able to help. $\endgroup$ – Gary Moore Oct 12 '15 at 10:00
  • $\begingroup$ @GitGud: Totally agree! I was being sloppy... $\endgroup$ – Gary Moore Oct 12 '15 at 10:00
  • $\begingroup$ @GitGud: And why not? If not you also cannot say things like "For any $c > 0$", for exactly the same reason, because "$c > 0$" is a sentence and in formal logic you can only quantify over the objects in the domain of discourse, not sentences.. $\endgroup$ – user21820 Oct 12 '15 at 14:58
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I see someone has already explained why not the options you listed. Alternative options, summing up comments :

  1. $\forall c\in\mathbb R^+_0\text\ \{0\}$
  2. $\forall c\in\mathbb R,c>0$
  3. $\forall c\in (0,\infty)$

From comments:

  1. $\forall c\in\mathbb R_{>0}$

(similar:this question)

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  • $\begingroup$ Thank you for a short and clear answer. $\endgroup$ – Danny_Kim Oct 12 '15 at 9:23
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    $\begingroup$ Don't mislead the OP, $\Bbb R_{0}^+$ denotes $[0,+\infty)$, not $(0,+\infty)$. $\endgroup$ – user236182 Oct 12 '15 at 20:12
  • $\begingroup$ For $(0,+\infty)$ (that is, only positive real numbers), you can use $\mathbb R^+$. $\endgroup$ – David K Oct 13 '15 at 19:32
  • $\begingroup$ @DavidK, I guess as explained in the article quoted at the end of the answer, it is unclear if $0\in\mathbb R^+$, which made me mention it as $\mathbb R^+_0\text\0$ $\endgroup$ – Jesse P Francis Oct 14 '15 at 4:46
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    $\begingroup$ @JessePFrancis But also note the opinion of tomasz. In any case, few people sift through all comments or follow all links, so it's good to make the answers as self-contained as they reasonably can be (without also making them too long). In any event, your answer is (and was) a nice collection of notations. $\endgroup$ – David K Oct 14 '15 at 11:59
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Maybe you can just put like this : $\forall c \in \mathbb{R}_*^+$.

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  • $\begingroup$ Are $\mathbb{R}_+^*$ and $\mathbb{R}_0^+$ same? $\endgroup$ – Danny_Kim Oct 12 '15 at 9:11
  • $\begingroup$ I'd take the "0" to mean "0 included" and the "*" to mean "0 excluded". $\endgroup$ – Gareth McCaughan Oct 12 '15 at 13:51
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    $\begingroup$ I would just use $\Bbb R^+$. $\endgroup$ – user236182 Oct 12 '15 at 20:15
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Another commonly used self-explanatory notation is $\mathbb{R}_{> c}$. Anything beyond half-line ranges would need some interval notation like $(a,b)$ or $]a,b[$.

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The formal way to write "for all $x$ such that $\phi(x)$ holds, $\psi(x)$ also holds" is something like the following:

$$ \forall x \,\phi(x) \implies\psi(x) $$

(I'm told in the comments that one needs to add parentheses around the implication, but I'm assuming the quantifiers have a lower precedence than material implication; your notation may vary)

You can then substitute $x \in \mathbb{R} \wedge x > 0$ for $\phi(x)$, giving us:

$$ \forall x \, (x \in \mathbb{R} \wedge x > 0) \implies\psi(x) $$

This is formally correct but not terribly readable, I'm afraid. In less formal contexts, you will want to use one of the other answers. But this is what you would write for automated theorem proving and other contexts where your notation has to be perfectly standard.

(Incidentally, this assumes you are working in classical logic, which has the notion of vacuous truth, that is, $\bot \implies \psi$ is a tautology for any $\psi$. In other logics, it gets messier and you have to specify what you mean by "such that." You can also break things by playing around with the domain of discourse (e.g. any statement about the reals is true if we aren't talking about the reals to begin with!) and so on, but this turns into metamathematical pedantry.)

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    $\begingroup$ Under the usual conventions for notation, your first formula means $(\forall x \phi(x))\Rightarrow\psi(x)$. But it should be $\forall x(\phi(x)\Rightarrow\psi(x))$. $\endgroup$ – David Oct 13 '15 at 4:51
  • $\begingroup$ @David: I've added a note about this, but I disagree with that parse. Quantifiers have low precedence. $\endgroup$ – Kevin Oct 13 '15 at 19:12
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    $\begingroup$ The convention I have always seen is that the scope of a quantifier is as short as it can possibly be (subject of course to forming a legitimate formula). $\endgroup$ – David Oct 13 '15 at 23:14
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    $\begingroup$ @David: That doesn't seem terribly useful to me. Generally, either the scope of the quantifier is an entire proposition, or it's inside parentheses. I don't see why you would give it such a high precedence. Regardless, some automated theorem provers use S-expressions and hence do not have this issue since everything is explicitly parenthesized anyway. $\endgroup$ – Kevin Oct 13 '15 at 23:15
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I've used "$\forall c \in \mathbb{R}_{>0}, \dots$", "$\forall c \in (0,\infty) \subseteq \mathbb{R}, \dots$", and "$\forall c \in \mathbb{R}, c > 0, \dots$". But I claim no normative authority.

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Taking advantage of the fact that a variable has a left side and a right side, and also, adding some parenthesis and brackets: can produce a logical and intuitive arguement: $$\forall(c:0<c\in\mathbb R )$$ I would normally just write $\forall(0<c\in \mathbb R)$.

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