9

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: $\forall c>0(c\in\mathbb R)$

Danny_Kim
  • 3,423
  • 3
    Some notations $\mathbb{R}^+_0$, $\left[0,\infty\right)$ – GAVD Oct 12 '15 at 08:46
  • 7
    $\forall c\in \mathbb R,c>0$. –  Oct 12 '15 at 08:48
  • Oh thank you. $\mathbb{R}_0^+$. Wow – Danny_Kim Oct 12 '15 at 09:10
  • @Danny_Kim: I suppose you refer to the Landau symbol at infinity? – C-star-W-star Oct 12 '15 at 10:23
  • $R^+$ is standard for the positive reals,as is $Q^+$ for the positive rationals.Or you can also write "$\forall c\in R ( c>0\implies S)$" to express "$S$ is true for every positive real $c$." – DanielWainfleet Oct 12 '15 at 10:28
  • @user254665 I won't count on 'standards' at such notations. Only the interval notation is unambiguous. – user251257 Oct 12 '15 at 11:03
  • 9
    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. – Gareth McCaughan Oct 12 '15 at 11:54
  • I would interpret $\mathbb{R}^+_0$ exactly like @GarethMcCaughan would, which is that it includes the zero. This other Math.SE question's answer indicates that $\mathbb{R}^+$ excludes zero, which suggests that $\mathbb{R}^+_0$ must somehow be different. [...] – hvd Oct 12 '15 at 13:40
  • [...] On the other hand, the comments there indicate that in some parts of the world, $\mathbb{R}^+$ or $\mathbb{R}+$ includes the zero. Wikipedia claims (do not trust this blindly -- research it yourself before relying on this information) that $\mathbb{R}+^*$ can be used to explicitly exclude zero in those countries. – hvd Oct 12 '15 at 13:40
  • @GarethMcCaughan: It's being used by many people (how many exactly I don't know) and I didn't notice your comment until after I wrote exactly that in my answer. – user21820 Oct 12 '15 at 14:59
  • Simply $\forall c\in \mathbb{R}^+$. –  Oct 12 '15 at 15:37
  • @251257 Kunen uses <x,y> for an ordered pair, perhaps because (x,y) can also mean an open interval. On this site I say "positive integer" explicitly because many members include 0 in the natural numbers,and many don't. As long as I can understand you, I don't mind what notation you use. Not that we don;t need standards. I'm not as concerned about it as some are.That's just me. – DanielWainfleet Oct 12 '15 at 19:48
  • 5
    @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. – tomasz Oct 12 '15 at 22:27
  • 1
    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. – tomasz Oct 12 '15 at 22:29
  • 1
    I guess what I mean by "isn't standard" is that it's not something I recall seeing used in a lot of books and papers. However, I've been out of the world of academic mathematics for some years now so (1) maybe my memory is deceiving me and (2) maybe the community has acquired better habits during those years :-). – Gareth McCaughan Oct 12 '15 at 23:57
  • Either as @YvesDaoust suggests, or just $\forall c>0$ if the context is clear that we are dealing with real numbers. – mickep Oct 14 '15 at 11:56

7 Answers7

14

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 $c > 0$".

Yes
  • 20,719
  • Thank you for informing to me about precise notation! – Danny_Kim Oct 12 '15 at 09:12
  • 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". – Git Gud Oct 12 '15 at 09:25
  • @Danny_Kim: Thank you; happy to be able to help. – Yes Oct 12 '15 at 10:00
  • @GitGud: Totally agree! I was being sloppy... – Yes Oct 12 '15 at 10:00
  • @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.. – user21820 Oct 12 '15 at 14:58
  • @user21820 Agreed. And I also avoid saying such things. I usually write "for any $c$ greater than $0$". – Git Gud Oct 12 '15 at 15:03
  • @GitGud: Oh wow at least you're consistent. I prefer the concise short-hand in most cases, since it is actually interpretable as valid English, but I never write things like "For any $a < c < b$", because English does not even support it. – user21820 Oct 12 '15 at 15:08
10

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)

Community
  • 1
Jesse P Francis
  • 1,469
  • 3
  • 27
  • 71
  • Thank you for a short and clear answer. – Danny_Kim Oct 12 '15 at 09:23
  • 5
    Don't mislead the OP, $\Bbb R_{0}^+$ denotes $[0,+\infty)$, not $(0,+\infty)$. – user236182 Oct 12 '15 at 20:12
  • For $(0,+\infty)$ (that is, only positive real numbers), you can use $\mathbb R^+$. – David K Oct 13 '15 at 19:32
  • @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 $\mathbb R^+_0\text\0$ – Jesse P Francis Oct 14 '15 at 04:46
  • @JessePFrancis As explained under the other question, the ambiguity is that some people consider $0$ positive (and also negative) whereas others do not. You could write $\mathbb R^+\setminus{0}$ unambiguously (where I think the brackets technically are required). This answer recommends $\mathbb R_{>0}$; is that good? – David K Oct 14 '15 at 11:43
  • @DavidK, Will add it as well! This answer is pointless now that we have quoted enough previous questions which deals with it in detail:-) Anyway, I guess I'll leave it here just for reference! – Jesse P Francis Oct 14 '15 at 11:48
  • @DavidK, also, quoting GarethMcCaughen from comments to the question "...$R_{>0}$, which so far as I know isn't standard but I think is clear and unambiguous." – Jesse P Francis Oct 14 '15 at 11:52
  • 1
    @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. – David K Oct 14 '15 at 11:59
2

Maybe you can just put like this : $\forall c \in \mathbb{R}_*^+$.

corindo
  • 3,752
2

Another commonly used self-explanatory notation is $\mathbb{R}_{> c}$. Anything beyond half-line ranges would need some interval notation like $(a,b)$ or $]a,b[$.

user21820
  • 57,693
  • 9
  • 98
  • 256
2

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.)

Kevin
  • 2,690
  • 1
    Under the usual conventions for notation, your first formula means $(\forall x \phi(x))\Rightarrow\psi(x)$. But it should be $\forall x(\phi(x)\Rightarrow\psi(x))$. – David Oct 13 '15 at 04:51
  • @David: I've added a note about this, but I disagree with that parse. Quantifiers have low precedence. – Kevin Oct 13 '15 at 19:12
  • 1
    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). – David Oct 13 '15 at 23:14
  • 1
    @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. – Kevin Oct 13 '15 at 23:15
0

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.

Eric Towers
  • 67,037
-1

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 $\forall(0<c\in \mathbb R)$.

Ranc
  • 1,977