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I want to express that a formula $Q(e)$ should hold for all elements $e$ of a set $S$ that fulfill some predicate $P(e)$.

These are the options that I know:

  • restrict the set:

$$ \forall e \in \{ e \in S | P(e) \} : Q(e) $$

This feels redundant with the duplicate "$ e \in $" parts.

  • use an implication:

$$\forall e \in S \: (P(e) \Rightarrow Q(e)) $$

This is concise, but it does not quite process in my mind the way I want to. I do not want the reader to think about the whole set, but the restricted set from the start.

  • use natural language:

"Let $e$ be an element of $S$ so that $P(e)$ holds. Then: $Q(e)$."

That works, but it is often not very convenient.

  • Also, this notation is common practise:

$$ \forall n < N: f(n) < f(N) $$

This is essentially already an abbreviated notation for a restriction to an upper bound.

But for restrictions other than to specific sets or to bounds, I cannot seem to remember how to notate them, even though I am sure that I have seen them many times, but I'm not able to find any examples.

I'm looking for a combination of all of the examples above, kind of like:

$$ \forall e \in S \cdot P(e) : Q(e) $$

For this, two kinds of 'separator symbols' are necessary, the first reading 'such that', the second 'the following should hold'.

This notation would have these main benefits:

  1. More clarity in the train of thought: All of the qualified statement is about the restricted set and nothing else
  2. The predicate $P(e)$ and the implication $Q(e)$ are both raised one syntactic level of nesting compared to the "implication" above, possibly eliminating the need for some parentheses,
  3. Rendering obsolete awkward mixed notations where it is unclear which variables are bound by the qualifier and which were present before, like this:

$$ \forall n < m \in \Bbb{N} : Q(n) \Rightarrow Q(m) $$

I thought I should add a concrete example, how about this:

$$\forall x, y, z \in \Bbb{R}^3 \cdot x = y + z : \lVert x \rVert_2 \le \lVert y \rVert_2 + \lVert z \rVert_2 $$

If this was written with an implication like this: $\forall x, y, z \in \Bbb{R}^3 (x = y + z \Rightarrow \lVert x \rVert_2 \le \lVert y \rVert_2 + \lVert z \rVert_2) $, I'd stop reading and think: "Huh, how is that condition ever going to just be randomly true." The notation above, on the other hand, indicates that the vectors are specifically chosen so that the predicate is true.

Is there a common notation for this? I'm not sure if the dot and colon combination is the way to go. I found nothing about this on this forum or other "trusty sources of knowledge". Thanks for any replies!

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What you're suggesting is certainly mathematically consistent. I definitely advise against inventing new notation—it will almost surely make things less clear for your readers.

Most readers of math would be quite comfortable with the implication $P(e)\implies Q(e)$ being equivalent to restricting the set of candidates. But if you really want to avoid this, just define a new set $T = \{e\in S\colon P(e)\}$ and then universally quantify over $T$.

I would also advise leaning towards natural language whenever possible. It's easiest for readers to understand, and part of developing our mathematical writing skills is structuring the surrounding text so that using natural language is actually convenient. (Clarity for the reader is more important than convenience for the author anyway.)

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