Correct notation for "for all positive real $c$"

Question

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

Answer 1

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 $c > 0$".

Answer 2

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

Answer 3

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:

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

(similar:this question)

Answer 4

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

Answer 5

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.)

Answer 6

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.

Answer 7

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