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?
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.
∧ is an operator because if P and Q are formulae, then so is (P)∧(Q). ∃x is an operator because if P is a formula then so is ∃x(P). ∃x∈R is an operator for the same reason. But ∃, by itself, is not an operator in this sense, so I don't think it should be declared as one.
– John Wickerson
May 22 '13 at 07:28
\colon is better than : when writing for example "For every x there exists y such that...".
– Jori Mäntysalo
May 22 '13 at 09:42
∃x is not a symbol by itself and so cannot be an operator in a typographical sense. The same is true for the integral: if f(x) is an formula, then \int f(x) is not a formula, but \int f(x)dx is. Yet, \int is a typographical operator. So \exists alone is not a logical operator, but \exists x\in M:P(x) is. Yet, \exists should be a typhographical operator.
– Toscho
May 22 '13 at 12:26
\let\oldexists\exists \let\exists\relax \DeclareMathOperator{\exists}{\oldexists} to continue writing \exists but get the above behaviour.
– gsvg
May 27 '17 at 11:31
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.
What about simple spaces "\ "?
– Gaston Burrull May 21 '13 at 19:37what do you think about use of "\ " or "," instead of "\ldotp"?
– Gaston Burrull May 21 '13 at 20:08
\ and , are fine alternatives. I incorporated , into my answer.
– John Wickerson
May 22 '13 at 07:42
Note that $\forall x \in \R. \ x^2 \geq 0$, because […]," as it's not immediately obvious whether the dot ends the sentence or the domain of quantification.
– wchargin
Feb 25 '17 at 20:02
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.
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.
About your suggestion, what about $\forall, c$? "," is a little space after quantifier also.
– Gaston Burrull May 21 '13 at 20:00\,, yes, it works (I used \mkern2mu to show how to adjust it). By the way the \quad if it's in a \displaymath I think it's much better than \ because it clearly separates the sentence.
– Manuel
May 21 '13 at 20:03
\quads as useful mathematical spaces. If I'm wrong, please correct me, it's true I know nothing about logic.
– Manuel
May 22 '13 at 10:02
:-)
– John Wickerson
May 22 '13 at 10:48
\end{ultrasexistcomment} Here if you swap the order the meaning changes whether we need to find any a,b for every c, d or we are looking for a particular pair of a,b that is valid for every c,d.
– percusse
May 22 '13 at 11:37
\quads.
– Manuel
May 22 '13 at 11:44
Another possibility is:
$\exists\ a,b \in \mathbb{R},\ \forall\ c, b \in\mathbb{R}$
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.
$\exists a\in\mathbb{R}$, $\exists b\in\mathbb{R}$, $\forall c\in\mathbb{R}$, and $\forall b\in\mathbb{R}$, or perhaps$\exists a, b \in\mathbb{R}$, $\forall c, d \in\mathbb{R}$. – Peter Grill May 21 '13 at 19:02$\exists a\in\mathbb{R}\ \exists b\in\mathbb{R}$can help. I agree with @percusse though. – marczellm May 21 '13 at 19:19\let\existstemp\exists \renewcommand*{\exists}{\existstemp\mkern1mu}(I don't exactly remember the space, I wrote the1muas an example). – Manuel May 21 '13 at 19:21\displaymath, e.g. in definitions, I think is clearer (and easy to remember) to write the symbols. – Manuel May 21 '13 at 19:23