# writing ''for all' with $\forall$

Suppose I want to write, in the context of a proof, say, that $x>0$ for all $x$ in some set $A$. There are several options how to do this:

1) $x>0$ for all $x\in A$
2) $x>0$ for all $x$ in $A$
3) $x>0$ for $\forall x\in A$
4) $x>0$ $\forall x\in A$

In my own writing, I usually opt for the third option based on the fact that it is shortest and uses fewest words (dominance to 1 and 2) but is more readable and 'looks more natural' than 4.

However, an argument can be made that 3 includes 'for' twice, one as a text and one in the $\forall$ symbol. (In fact, a referee recently pointed me to a 'typo' when I used the third formulation.)

Any thoughts/guidelines/preferences/other suggestions on style connected to $\forall$?

• "for $\forall$" looks weird to me precisely because I read it as "for for all". I go with $1$ or $4$ depending on the audience. – user137731 Jan 11 '16 at 5:36
• ??? the 1st order logic way to write it is definitely $\forall x \in A, x > 0$ en.wikipedia.org/wiki/First-order_logic – reuns Jan 11 '16 at 5:41
• (2) is correct, but I'd also accept (1). (4) which you see a lot here, is atrocious, and (3) is even worse. Symbols like $\forall$ and $\exists$ should be used if you're writing formulas in first order logic; they have no place in an English sentence. – bof Jan 11 '16 at 5:42
• It's generally considered bad form to use abbreviations like $\forall$ in formal papers, apart from explicit formal expressions; the English equivalent is cleaner and easier to read (especially when there are multiple quantifiers). If it's for your own notes or something informal, it really doesn't matter what convention you adopt. Also, I don't see any reason to prefer item (2) over (1). – anomaly Jan 11 '16 at 5:54
• I'm surprised there's so much love for (2) here; I consider (1) to be significantly better than (2). I'm not sure why; it just looks more natural. I think part of the reason is that "in" can have many different meanings, and even if the meaning is clear from context, it is always nice to give a visual cue that you should be thinking about set elements. And in contrast with $\forall$, $\in$ is not any more difficult to read than "in". – Eric Wofsey Jan 11 '16 at 6:00

## 4 Answers

Formally, I have usually seen $\forall x\in A (x \gt 0)$ (the parentheses vary) or $x \in A \implies x \gt 0$ depending on the language you are using. Informally I prefer putting the quantifier first because it tells you the range of $x$ that are of interest, then tells you what is interesting about $x$, but I can accept all of them.

When I was in school, most professors used the first option when they were giving lectures. While they were writing the theorems, lemmas, etc on the board, I feel that that particular style (first option) made the lecture easier to follow - especially when you don't have much experience using mathematical notation like that.

However, in more advanced math classes, professors would usually follow the fourth option when proving concepts and giving definitions. Also, in the more advanced classes I took, I also followed the fourth option when doing homework and taking exams. Since I have no experience teaching math formally, I cannot say what is the correct thing to do. All I can say is, use your best judgement and know your audience. Also, I would say do not use option three. It doesn't make much sense when you read it aloud.

I wouldn't recommend using any of them. Something simple and to the point is probably best: $$(\forall x\in A)(x>0).$$ To hammer this point home, imagine trying to do what you are doing with multiple quantifiers; that is, things will get ugly if you simply try to lump everything together and do not list the quantifiers first.

For example, imagine you are trying to communicate that "$f$ is a constant function, where $x$ and $y$ are real numbers." How would you write this using quantifiers? Surely not $$f(x)=y\;\exists y\in\mathbb{R}\;\forall x\in\mathbb{R}.$$ That looks horrible. A far more elegant and clearer alternative (where it is assumed at the outset that your domain of discourse is the set of real numbers): $$(\exists y)(\forall x)(f(x)=y).$$

• Would love to hear feedback as to why someone downvoted this. – interrogative Jan 11 '16 at 5:50
• You may be right, but this is not at all the mathematical convention. Generally, it is expected that things be written in English, like "$x > 0$ for all $x \in A$", in mathematical writing. Also, your example is weird, because I have no idea why $y$ is paired with $\exists$ and $x$ is paired with $\forall$. (I didn't downvote.) – 6005 Jan 11 '16 at 5:53
• I think this answer is stated far too absolutely; everything depends on context. Yes, putting the quantifiers afterwards can get unwieldy in some contexts, but in others it can sound perfectly natural. Also, in any reasonably formal mathematical writing, it is usually preferable to avoid quantifier symbols as they tend to be harder to read than English words. – Eric Wofsey Jan 11 '16 at 5:55
• @6005 What is the mathematical convention? To express things in English? Sure, but OP wants to use quantifiers. In such a case, I think what I provided is clear and more understandable (of course, people are free to disagree). The example I gave is simple: to describe a constant function $f$, we can simply say that "there exists a real number $y$ for all real numbers $x$ such that $f(x)=y$." That is all that was meant. – interrogative Jan 11 '16 at 5:55
• @interrogative Yes, to express things in English--OP seems to be asking about writing a proof to a mathematical audience, not a formal proof, though I could be wrong. I did misunderstand your example, which makes perfect sense to me now. The part that confused me is "f is a constant function, where x and y are real numbers", which should really just read "f is a constant function" for your post to be technically correct. – 6005 Jan 11 '16 at 5:58

The normal way is to use the $\forall$ in prefix form, that is something like $\forall x\in A: \phi(x)$. There's a number of ways this is done (fx skipping the colon, putting $\forall x\in A$ in parentheses or the statement in parenthesis), but generally they're all understood.

The key part here is that it should be readily understood by the reader (or you have to have a legend with the notation you use). I can hardly see that any of the notations you gave being misunderstood or not readily understood. Generally you shouldn't bother using a few extra tokens if that improves understandability, therefore probably the former is better than the later.

You should note that using postfix notation you would probably reverse the order of the quantifiers - since the quantifier applies to the entire statement preceding it. That is $\forall x\exists y\phi(x,y)$ should in postfix notation become $\phi(x,y)\exists y\forall x$.