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The following are variations in expressing "there exists a natural number $n$, such that $2T = n \cdot (n+1)$":

  • $\exists n\in \mathbb {N},\; 2T = n \cdot (n+1)$
  • $\exists n\in \mathbb {N}\; [2T = n \cdot (n+1)]$
  • $\exists n\in \mathbb {N}\; {2T = n \cdot (n+1)}$
  • $(\exists n\in \mathbb {N})\; 2T = n \cdot (n+1)$
  • $\exists n\in \mathbb {N} \; : \; \{ 2T = n \cdot (n+1) \}$
  • $(\exists n\in \mathbb {N}) \; (2T = n \cdot (n+1) )$
  • etc

My question is: what is the official, or recommended, notation?

Does the answer depend on UK-USA or other traditions?

I have seen variations in many respected courses and textbooks.

Attached is Pro Keith Devlin's usage from his "Introduction to Mathematical Thinking" course - you can see he uses plenty of round and square brackets - but not colon which I had always used for "such that".

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This is a related question but doesn't answer this question which asks for official / recommended / tradition: How do commas and brackets affect the meaning of quantifiers?

Tankut Beygu
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Penelope
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    I doubt there is a universally recommended tradition. I've seen many of the above and never thought any one was violating some standard convention. There certainly isn't anything official because there's no one to officiate. – Ethan Bolker Nov 22 '24 at 20:21
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    I don't like the third option because it might be difficult to figure out the end of the quantifier and the start of the dependent clause for more complicated expressions if you don't have any deliminators. But apart from that, pretty much everything is fine. – anankElpis Nov 22 '24 at 20:31
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    I've often said things like "do whatever you want but do it consistently." But here, I'd weaken "consistently" because it's reasonable to use parentheses around long expressions but not short ones. The depth of nesting can affect what punctuation (parentheses, brackets) goes right after the quantifier. Also, I really dislike any form of "such that" (the words or "s.t." or a colon) after the existential quantifier The notation $\exists n$ means "there exists $n$ such that" or "for some $n$." I think $\exists$ and $\forall$ should use the same syntax. – Andreas Blass Nov 22 '24 at 20:37
  • Yes, ∃x s.t. (Px) is an eyesore, lying in no man's land between the verbal and the symbolic. I'm not a fan of ∃x: P(x) (or ∀x, P(x) or ∀x: P(x) or ∃x, P(x)) either, because they invariably make me wonder (or puase to decide) whether to read into the punctuation. – ryang Nov 23 '24 at 09:46
  • No official one. As per linked post, are all tiypographical variants, someone more clear than others. – Mauro ALLEGRANZA Nov 23 '24 at 09:54
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    The style of parentheses is irrelevant. The only issue is to clearly identify the scope of the quantifier: the formula to which the quantifier applies. It is crucial for complex formulas (not the case above). – Mauro ALLEGRANZA Nov 23 '24 at 10:00
  • I think Mauro, your point about scope is concrete and important. Thank you. – Penelope Nov 23 '24 at 13:14

1 Answers1

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Long comment

The style of parentheses is irrelevant: the only issue is to clearly identify the scope of the quantifier: the formula to which the quantifier applies.

It is crucial for complex formulas, in order to manage the difference between e.g. $\forall x (Px \land Qx)$ and $\forall xPx \land Qx$.

In your case, formula $2T=n⋅(n+1)$ is atomic (has no sentential connectives) and thus the simple form $∃n \in \mathbb N,2T=n⋅(n+1)$ or $∃n \in \mathbb N (2T=n⋅(n+1))$ will be enough.

Mauro ALLEGRANZA
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