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What is the best way to denote it?

  1. $\forall i\in\{1,\dots,n\}: P(i)$;
  2. $\forall i=1,\dots,n: P(i)$;
  3. $\forall i=\overline{1,n}: P(i)$;
  4. $P(i)$ for $i=1,\dots,n$;
  5. ...

2 Answers 2

11

Use "$P(i)$ for $i=1,...,n$" to refer to the property being true in those cases. Using existential and universal quantifiers in ordinary mathematical prose (as opposed to formal logic text) is ugly.

  • 0
    Some mathematicians disagree with e$x$cluding quantifies. However there are theorems about logic of quantifiers and some definitions include a long combination of quantifiers (the$y$ are frequent in anal$y$sis).2013-02-12
1

I don't know if I've ever seen the notation $\overline{1, n}$, so I wouldn't use that :) But, maybe others know it, so maybe it's okay. Other than that, I wouldn't use $\forall$ or $:$ ever and most of my experience with such notation is in the class where you first learn it. I would use words but the basic forms of your 1, 2, and 4 seem pretty good.

$P(i)$ is true for all $i$ such that $1 \leq i \leq n$.

$P(i)$ is true for $i = 1, \ldots, n$.

For $i = 1, \ldots, n$, $P(i)$ is true.