When to use universal quantifier, for and when not?

Question

Say, I want to express that the probability that a discrete random variable takes on some specific values needs to be greater zero. This is clear if all natural numbers are meant: $$\mathbb{P}(X=x)>0,\forall x\in\mathbb{N}$$

Now, if I am only interested in, say, $0\leq x\leq 5$. I see three possibilities:

1)

$$\mathbb{P}(X=x)>0,\forall x(0\leq x\leq 5)$$

2) $$\mathbb{P}(X=x)>0,\text{for }0\leq x\leq 5$$

3) $$\mathbb{P}(X=x)>0,0\leq x\leq 5$$

Which of these possibilities are valid and which are not and why or why not? Also, are the valid ones equivalent? And could I replace "," with ":" at some point?

Answer 1

If we work with a fully formalized syntax, like first-order logic, we have to write :

$$∀x[(x ∈ \mathbb N ∧ (0 ≤ x ≤ 5)) → (\mathbb P(X=x)>0)].$$

But, IMO, there is no need (and no real benefit) to use fully formalized expressions in "usual" mathematical contexts.

Thus, the issue is only about readibility, and so the three are "equivalent": maybe 2) is a little bit less cumbersome.

Answer 2

The comma makes$\color\red,$ sense only in (3); it's like the red comma that I've just randomly inserted, which interrupts the sentence's flow. A colon in lieu of it makes equally little sense.

A few suggestions in ascending order of formality (the most formal is in Mauro's answer):

• $\mathbb{P}(X=x)>0\quad(x=0,1,\ldots,5)$

• $\text{For each natural $x$ such that }0\le x\le5,\; \mathbb{P}(X=x)>0$

• $\forall x{\in}\big\{x\in\mathbb N\mid0\le x\le5\big\}\,\; \mathbb{P}(X=x)>0$

• $∀x{∈}\mathbb N\; \Big(0\le x\le5 \implies \mathbb{P}(X=x)>0 \Big)$