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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

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

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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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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:

- Avoid any unnecessary symbols like “:” or “.”
- 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 $\exists x \forall y \varphi(x,y)$.
- 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)

$$ - 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.

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