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My question is about the correct way to write such statements like $$\exists y \forall x f(x) \leq y,$$ which means "there exists an $y$, such that for all $x$, $f(x)$ is less than or equal to $y$."

I have seen variations using round and square brackets, but I don't think I have ever read anything that states one version is the correct or official standard:

$$(\exists y \forall x) \;f(x) \leq y$$ $$(\exists y \forall x) \;[f(x) \leq y]$$ $$\exists y \forall x \;[f(x) \leq y]$$ $$[\exists y \forall x] \;(f(x) \leq y)$$

Things get harder to read if more information is required, for example:

$$(\exists y \in \mathbb{N}, \forall x \in \{i \in \mathbb{N}:1 \leq i \leq n\}) \; [f(x) \leq y].$$

Should that comma be there?

I'd value answers which cover:

  • historical evolution and regional traditions of the notation
  • square vs round brackets and where to use them
  • how best to add the additional information without overloading the reader
ryang
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Penelope
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    Why would you need brackets ? – Iq-n-dI Jun 07 '24 at 23:49
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    May I propose $$\exists y; [\forall x ;f(x) \leq y]$$ ? – peterwhy Jun 07 '24 at 23:51
  • @peterwhy you may propose but the question is asking about official standards for notation, or widely adopted convention. Also, I have never seen one quantifier outside brackets containing another - it reads well, but I have to ask why I have never seen it in traditional textbooks? – Penelope Jun 08 '24 at 00:16
  • @Maxime because for more complex statements with multiple quantified variables, brackets makes it easier to read. But are you suggesting the official standard is not to use any brackets? – Penelope Jun 08 '24 at 00:17
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    Yes, I am, and you’d put brackets de delimit “blocs of symbols” that form properties if you really want to. For exemple if P and Q are polynomials $\exists x\exists y,; P(x,y) = 0\wedge Q(x,y)=0$ would be transformed in $\exists x\exists y,; ((P(x,y) = 0)\wedge( Q(x,y)=0))$ – Iq-n-dI Jun 08 '24 at 05:23

1 Answers1

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There is no "official standard" or singular "correct way".

These logical symbols are taught to undergraduates (say) as a tool to help them learn how mathematical logic works. There are thousands of different textbooks and curricula for teaching logic, and they probably use dozens of different conventions concerning brackets, commas, and so on.

When it comes to research-level mathematics, where communication of complicated mathematical statements is crucial, best practice is to not use those symbols for quantifiers. Instead, we write complete sentences in our language of choice; some of the nouns and verbs are made of mathematical symbols rather than letters, but words are the way we convey the relationship between the mathematical pieces.

With your example at the end, I would write it as $$ \text{"There exists $y\in\Bbb N$ such that $f(x)\le y$ for all $x\in\{1,\dots,n\}$."} $$ (If it were clear from the context that $x$ must be an integer, we could also write "for all $1\le x\le n$". To be honest, I would seriously consider starting the sentence "There exists a positive integer $y$ such that" as well.)

The goal of mathematical writing is not to convert as much as possible into mathematical symbols. The goal of mathematical writing is to be as clear as possible while remaining precise. Mathematical symbols are a tool to be used towards that end, not an end in and of themselves.

Greg Martin
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  • I agree with everything said, however for absolute clarity I'd eliminate the hanging quantifier by writing There exists y∈N such that for all x∈{1,…,n} f(x)≤y instead of There exists y∈N such that f(x)≤y for all x∈{1,…,n}. $\quad$ 2. The first comma in the OP's there exists an y, such that for all x, f(x) is less than or equal to y (symbolically: $\exists y {\in} \mathbb{N}; \forall x {\in} {1,\ldots,n} ; f(x) \leq y$) is really jarring and shouldn't be there, since it is followed by a restrictive clause and not signalling a pause.
    • – ryang Jun 08 '24 at 07:16
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    thank you both Greg and ryang - both very informative replies. As a self-teaching I am actually surprised that professionals will sue English sentences over symbols. I had expected them to avoid human natural language as being subject to too much imprecision and ambiguity vs the more rigidly defined and restricted symbolic formulae. – Penelope Jun 08 '24 at 11:43
  • @Penelope To add: "Should that comma be there in the formalisation?" In my opinion, no, because in formal logic superfluous punctuation potentially introduces ambiguity (though not in your given example) and decreases readability. – ryang Jun 10 '24 at 02:32
  • @ryang "Hanging quantifier" is, in my opinion, a vestige of undergraduate formalism. When the individual clauses are simple as in this case, I don't think any reader would be misled as to how the quantifiers are operating. – Greg Martin Jun 10 '24 at 06:30
  • Not arguing at all against stylistic choices or being prosaic; just suggesting that the habit of consistently avoiding hanging quantifiers has value against potential ambiguity, imagined or otherwise. Apologies for cluttering here! – ryang Jun 10 '24 at 07:14