Best practice for typesetting quantifiers?

Question

When I work with quantifiers I noted that are very close to the other symbols and the result does not look good, for example

$\exists a\in\mathbb{R}\exists b\in\mathbb{R}\forall c\in\mathbb{R}\forall d\in\mathbb{R}$

Which is the proper form to write quantifiers?

Answer 1

It depends on the context.

If this is part of a piece of text, then you might consider Peter Grill's suggestion:

$\exists a\in\mathbb{R}$, $\exists b\in\mathbb{R}$, 
$\forall c\in\mathbb{R}$, and $\forall b\in\mathbb{R}$

On the other hand, if the quantifiers are part of a logical formula, you might consider a dot between the quantifiers, like this:

$\exists a\in\mathbb{R}\ldotp\exists b\in\mathbb{R}\ldotp
\forall c\in\mathbb{R}\ldotp\forall b\in\mathbb{R}\ldotp P$

This dot notation is inherited, I think, from Russell and Whitehead's Principia Mathematica, and is quite widely used, particularly in computer science. A comma between quantifiers is quite unusual, though it does appear in the syntax of the Coq theorem prover.

$\exists a\in\mathbb{R}, \exists b\in\mathbb{R}, 
\forall c\in\mathbb{R}, \forall d\in\mathbb{R}, P$

The comma notation becomes awkward when you want to quantify several variables at the same time, because then you have two different types of comma in the same formula:

$\exists a,b\in\mathbb{R}, \forall c,d\in\mathbb{R}, P$

In such cases, you might consider putting just a space between the variables, like this:

$\exists a\;b\in\mathbb{R}, \forall c\;d\in\mathbb{R}, P$

The idea of putting spaces between variables, rather than commas, is taken from the syntax of the Isabelle theorem prover.

Answer 2

Simply make these characters what they should be: Operators. They aren't arithmetic operators but logical ones, but that doesn't make any difference here:

\documentclass{article}
\usepackage{amsmath,amssymb}
\DeclareMathOperator{\Exists}{\exists}
\DeclareMathOperator{\Forall}{\forall}
\begin{document}
$\Exists a\in\mathbb{R}\Exists b\in\mathbb{R}\Forall c\in\mathbb{R}\Forall d\in\mathbb{R}$

$\Exists a\in\mathbb{R}:\Exists b\in\mathbb{R}:\Forall c\in\mathbb{R}:\Forall d\in\mathbb{R}$

$\Exists a,b\in\mathbb{R}:\Forall c,d\in\mathbb{R}$
\end{document}

Additionally, I would add a colon which stands for "such that".

Last but not least, it's equivalent but easier to grasp, if the both "exists" and "foralls" are grouped. R^2 would be wrong in this case, because a and b should each be in R. (a,b) would be in R^2, but that's not written.

Answer 3

In my opinion, the real issue with quantifiers is that it's hard to obtain consistent spacing, as I explained in this answer. The most striking example I found: \[\forall W\forall A\] gives

Of course there should be more space before the second quantifier; a single space \   will usually be OK. The problem is the spacing after the quantifiers. There is no simple solution to this, other than using manual kerning where needed. In this case, \[\forall\mkern2mu W\ \forall\mkern-1mu A\] looks quite alright:

Let me point out that I'd use quantifiers only in displayed formulas, never in inline math.

Answer 4

I don't know if this is what you are asking, but it's related.

In my opinion it's horrible the space after the quantifiers (they look very close to the next letter). I always edit them and add an small space

\let\existstemp\exists
\let\foralltemp\forall
\renewcommand*{\exists}{\existstemp\mkern2mu}
\renewcommand*{\forall}{\foralltemp\mkern2mu}

By the way, as others are saying, it depends on the situation. If it's inline I would go for There exist real scalars a,b for all real scalars c,d (Percusse's comment). But if it's inside a \displaymath I would go for the symbols.

First of all, I usually space my math with \quads (this is personal taste, and you have to choose what you use). And, in second place, I don't know how your example should be read:

• If it's read There exist real scalars a,b for all real scalars c,d I would change the order and write For all real scalars c,d there exist real scalars a,b… and write \forall c,d \in \mathbb{R} \quad \exists a,b \in \mathbb{R}.

• And if it's read as There exist real scalars a,b such that for all real scalars c,d… then I would write \exists a,b \in \mathbb{R}, \quad \forall c,d \in \mathbb{R}

Here it is a full example.

\documentclass{article}
\usepackage{amssymb}

\let\existstemp\exists
\let\foralltemp\forall

\begin{document}
\[
    \exists a,b \in \mathbb{R}, \quad \forall c,d \in \mathbb{R}
\]

\renewcommand*{\exists}{\existstemp\mkern2mu}
\renewcommand*{\forall}{\foralltemp\mkern2mu}

\[
    \exists a,b \in \mathbb{R}, \quad \forall c,d \in \mathbb{R}
\]
\[
    \forall c,d \in \mathbb{R} \quad \exists a,b \in \mathbb{R}
\]
\end{document}

In order to justify the \quads instead of the \s, here is another example which, in my opinion, shows my idea (and why in displaymaths \quads are useful):

I think that the first line is far more readable than the second one.

Answer 5

Another possibility is:

$\exists\ a,b \in \mathbb{R},\ \forall\ c, b \in\mathbb{R}$

Answer 6

I've always used \; after every symbol that goes with a quantifier. For example,

\begin{equation*}
\forall \varepsilon > 0 \; \exists N \in \mathbb{N} \; \forall n \in \mathbb{N} \; 
(n \geq N \implies |s_n - L| < \varepsilon)
\end{equation*}

Though I do understand that such an ad hoc method is not good practice.