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We all know that there are different ways to say that e.g. an element $x$ belongs to each member of a family of sets $(A_j)_{j \in J}$ for some index set $J$. The most common ways I know are the following:

  • $\forall j \in J \colon x \in A_j$
  • $x \in A_j$ for all $j \in J$
  • $x \in A_j \ \forall j \in J$

I think I heared that from some people's point of view the last notation is not considered as "nice". Now my question to the pros outside there is: Is it true that one should not write the universal quantifier behind a statement? Do you recommend using one of the first two notations only? I think there are even some textbooks using the last notation so I am not sure if it is seen as "bad" my most mathematicians. Maybe I am just influenced by some special opinions.

Rodrigo de Azevedo
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math_space
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  • I use the second one most of the time. Only in a formal setting I would use the first one, or actually $\forall j \in J. x \in A_j$, or even $(\forall j \in J), x \in A_j$. I hardly ever see the third one, and never use it. – Luca Bressan Oct 08 '16 at 17:48
  • The problem comes when there are multiple quantifiers involved. For instance, say you have the statement "$\forall x\in X, \exists \delta>0$ such that $B(x,\delta)\subseteq X$" If you write the universal quantifier last, "$\exists \delta>0$ such that $B(x, \delta)\subseteq X \ \forall x \in X$," this can be misconstrued as meaning "$\exists \delta>0$ such that $\forall x \in X, B(x,\delta)\subseteq X$." Which has a different meaning than the original sentence. – kccu Oct 08 '16 at 18:03
  • Yes, this is clear of course. I am only speaking about examples similar to the one I presented. – math_space Oct 08 '16 at 18:12
  • My answer is here: hanging quantifiers. – ryang Feb 24 '22 at 16:40

2 Answers2

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If you use formal notation, you should use it correctly. The rule for $\forall$ and $\exists$ is that they are written to the left of the formula they apply to, just like $\sum$, $\int$ and $\lim$ do; nobody would dream of writing the latter after the formula they apply to. The only reason people would write $\forall$ after the formula is that it is pronounced "for all" and in English a phrase starting with "for all" may come after the statement it applies to (although it might come before as well). I have not seen the same thing happening with $\exists$, and I think this is because in in English a phrase that would start with "there exists an$~x$ such that" when placed before the statement would become "for some$~x$" when placed after the statement, so one cannot place $\exists$ to the right and read it as a simple abbreviation. But neither $\exists$ nor $\forall$ are abbreviations for an English phrase, and they should not be used as such.

Marc van Leeuwen
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It depends whether you are writing on a blackboard or in a formal article. In a formal article, I would write

$x$ belongs to $A_j$ for all $j \in J$

to have a fluent sentence. See Halmos' recommendations on How to write mathematics. On a blackboard, I may simply write

$x \in \bigcap_{j\in J} A_j$

J.-E. Pin
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