For years I have been typing and writing quantifiers in a certain way. Now that I am writing my thesis, my adviser is taking issue with some of these things. Since he is my adviser I'm going to do what he says, but I am curious about the general consensus on this.

As an example, let's say I wanted to write symbolically "There exists an element $a$ of $A$ such that $a$ is positive." My habitual way of doing this would be

$$\exists a\in A: a>0.$$

My adviser has 2 problems with this. Firstly he says there should be a space between the $\exists$ and the $a$. Secondly he says I cannot assume people will read the colon as "such that." So he would have me change this to:

$$\exists\ a\in A\text{ such that }a>0.$$

Which seems correct to you?

As for the space after the $\exists$, it looks funny to me. It also seems significant that $\LaTeX$ does not automatically put a space after the $\exists$ and I need to write \exists\ instead of \exists.

As for the colon, it's been a few years but I used to study logic, and I think in the conventions there my usage is fine. I'm not sure of the grammatical terminology but there is a sense in which the colon indicates that we are done quantifying things and are now going to indicate the property the quantified things have. Not only have I been writing this way for years, I have been teaching students to write this way.

I was never quite sure though about the colon with a universal quantifier? Like, is it conventional at least for some people, if I write "For all elements $a$ of $A$, $f(a)=c$" as follows?

$$\forall a\in A: f(a)=c.$$

A lot of people don't write the universal quantifier at the beginning either. I feel that my more hardcore logic professors would never do this, but a topologist would have no problem writing:

$$f(a)=c, \forall a\in A.$$

The predicate logic conventions, if I'm remembering them properly, seem just way more ... logical. But I want my writing to be familiar to my audience, which is something my adviser has the best feel for. I guess once I am more established I will have more freedom in how I write. In the meantime I'd like to hear which of these things look correct or incorrect, and please also mention what area of math you work in because that seems to matter.

UPDATE: I appreciate the many insightful comments this question has received, but would someone please post their answer as an answer?

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    $\begingroup$ @lhf it is not a question about how to do anything in tex, but a question about what is the correct mathematical notation/convention. I think it is more appropriate as a question here, tagged "notation." $\endgroup$ – j0equ1nn Feb 26 '16 at 1:22
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    $\begingroup$ Every textbook in formal logic has its own set of conventions, but there are some commonalities. The colon is not extremely common. The lack of space around quantifiers works well when there are parentheses: $(\forall x)(\exists b)\phi(a,b)$. When there are no parentheses, the default spacing does often look tight: $\forall a\exists b\, \phi(a,b)$. But no notation system is any more "correct" than any other; as long as they are unambiguous, they are all equally correct, even if some are more common than others. $\endgroup$ – Carl Mummert Feb 26 '16 at 1:25
  • $\begingroup$ I find it strange to write out "such that." I like the space, but that's only preference, it doesn't change the meaning. I feel using a colon is quite common and is what I would write. $\endgroup$ – Carser Feb 26 '16 at 1:26
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    $\begingroup$ I find the use of a colon to separate a quantifier from its matrix most unusual (except in set comprehensions, where I personally would use $\mid$), but I have been seeing it a lot recently on MSE. Where did it come from? My advice to the OP would be to find a book or paper with well thought-out notation that you find acceptable and systematically follow that notation. $\endgroup$ – Rob Arthan Feb 26 '16 at 1:35
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    $\begingroup$ I would not use either version. My preference is for $\exists a\in A\,(a>0)$. (I’m a set-theoretic topologist.) $\endgroup$ – Brian M. Scott Feb 27 '16 at 0:19

First of all, I would avoid logical symbols as much as possible. For instance, you initial example could be phrased as "There exists a positive element $a$ in $A$". Now, if you really need to use quantifiers, my advice would be the following:

  1. Avoid any unnecessary symbols like ":" or "."
  2. Don't hesitate to add spaces, and possibly parentheses and brackets to improve readability. $\LaTeX$ offers a large panel of possibilities to do so. For instance, $\exists x\ \forall y\ \varphi(x,y)$ looks better than $\exists x \forall y \varphi(x,y)$.
  3. Put quantifiers in the front, not at the end. Although it is acceptable to write "$f(n) > 0$ holds for every integer $n$", if you really need to use quantifiers, it is preferable to write $$ \forall n \in \mathbb{Z}\quad f(n) > 0 $$ or, as suggested by Brian M. Scott, $$ \forall n \in \mathbb{Z}\quad (f(n) > 0) $$
  4. Double check again. Do you really need quantifiers? You will not find a single quantifier in Rudin's Real and Complex Analysis. You will no find quantifiers in Bourbaki's chapters on topology either, although Bourbaki's style is usually very formal.
  • $\begingroup$ Thanks, and I think you make good points here. It is true that quantifiers could be avoided. Personally I find math easier to read in symbols than in English, wherever possible, but on the other hand I am not just writing things for myself to read (I hope). I absolutely agree about quantifiers belonging at the beginning, I just wish more people were careful about this (even when writing on a blackboard). $\endgroup$ – j0equ1nn Mar 14 '16 at 20:00

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