(Some report that my question is similar to another post. However, that post is talking about writing the "proof", rather than "stating" the theorem. "Proving" a theorem is NOT of the same structure and situation as "stating" a theorem. So this question is not duplicated to the other! Do not let it to be closed! And by the way, I'm also the OP of that question...)
In writing a textbook, when we need to state a theorem that is a universal quantification, we can use the word
"for all ..."(or equivalently "for every", "for any", "for arbitrary", "for each")
or
"let ...",
Which of these ways is more ideal? Why?
Although I think writing as "for all" is the more natural way to reflect the logical structure, that is a universal quantifier $\forall$, the popular style I have seen tends to use "let".
Any theoretical aspect or experience is welcome.
Example set 1.
For all natural numbers $n$, if $n$ is even, then $n$ squared is even.
Let $n$ be a natural number. If $n$ is even, then $n$ squared is even.
Example set 2.
Let $A,B$ be two sets. If for all $x\in A$, $x\in B$, then we say $A$ is a subset of $B$.
For all pairs $A,B$ of sets, if for all $x\in A$, $x\in B$, then we say $A$ is a subset of $B$.
Example set 3.
Let $Y$ be a subspace of $X$. Then $Y$ is compact if and only if every covering of $Y$ by sets open in $X$ contains a finite subcollection covering $Y$. (Munkres Topology Lemma 26.1)
For all subspaces $Y$ of $X$, $Y$ is compact if and only if every covering of $Y$ by sets open in $X$ contains a finite subcollection covering $Y$.
Example set 4.
- For every $f:X\to Y$ being a bijective continuous function, if $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism. (adapted by me, maybe ill-grammared?)
- For every bijective continuous function $f:X\to Y$, if $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism. (adapted by me.)
- Let $f:X\to Y$ be a bijective continuous function. If $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism. (Munkres Topology Theorem 26.6)
New added example set 5(I skipped the quantification on $E,f:E\to\mathbb{R},L,c$, just focus on the key part here.)
$\endgroup$ 16
- If "$\forall\varepsilon>0,\exists\delta>0,\forall x\in E,0<|x-c|<\delta\rightarrow |f(x)-L|<\varepsilon$", then we say $f(x)$ converges to $L$ when $x$ approaches $c$.
- If, for all $\varepsilon>0$, there exists $\delta>0$ such that for all $x\in E$, if $0<|x-c|<\delta$ then $|f(x)-L|<\varepsilon$", then we say $f(x)$ converges to $L$ when $x$ approaches $c$. (Using "for all")
- If let $\varepsilon>0$, there is $\delta>0$, such that let $x\in E$, if $0<|x-c|<\delta$ then $|f(x)-L|<\varepsilon$", then we call $f(x)$ converges to $L$ when $x$ approaches $c$. (Using "let". I think this type is not natural. But I can't tell why.)
6 Answers
$\begingroup$If you're doing informal mathematics, there's really no difference. I guess from a type-theoretic perspective, it's kind of the difference between $$x:\mathbb{R} \vdash P(x) \qquad \mbox{and} \qquad \vdash (\forall x:\mathbb{R})\,P(x).$$
The former is arguably better, because it doesn't presuppose that we're working in a category that interprets universal quantification. So "let" is preferable to "for all" for this reason. But, again, unless you're doing highly formal mathematics, it's not really worth worrying about. (I say that, but a part of me finds the question fascinating, and I've just gone and favourited it.)
$\endgroup$ 12 $\begingroup$In your first example, either is fine. In your second example, 2 is ungrammatical - you cannot just replace "Let" with "For all", the "be" has to be deleted (or replaced with "that are"). Otherwise, either is fine.
In all three examples, you'll notice that your "For all" example results in a long, slightly awkwardly-phrased sentence with at least three clauses. The "Let" version divides the sentence into two simpler sentences, so the reader can process it one step at a time.
In general, "For all" is okay as long as the thing you're quantifying over is small and doesn't really require any work to understand. If the reader's going to have to think about it even a little - e.g. "system of linear equations in five variables" - I'd use "Let".
Also - and I'm pretty sure this is just a personal preference - I try to avoid having more than two parts to a sentence in a mathematical theorem, if I can. "If $X$ then $Y$" is fine, but "If $X$ then if $Y$ then $Z$" is ugly. Similarly, "For all $x$, if $Y$ then $Z$" is complicated, and it gets worse the more complicated $x$, $Y$, and $Z$ are.
$\endgroup$ 2 $\begingroup$As a personal point of view I would first "fix" what I'm working with with a let, and then state my property with for all if needed.
So my canonical form would be:
Let a be something, b be something, and c be something. If for all d such that something on a,b,c and d then we have something great on a,b,c
I think it is the more readable way to state a theorem.
So to go through your examples:
Example set 1.
Let $n$ be a natural number. If $n$ is even, then $n$ squared is even.
Example set 2.
Let $A,B$ be two sets. If for all $x\in A$, we have $x\in B$, then we say that $A$ is a subset of $B$.
Example set 3.
Let $Y$ be a subspace of $X$. $Y$ is compact if and only if every covering of $Y$ by sets open in $X$ contains a finite subcollection covering $Y$. (Munkres Topology Lemma 26.1)
Example set 4.
Let $f:X\to Y$ be a bijective continuous function. If $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism. (Munkres Topology Theorem 26.6)
** example set 5**
$\endgroup$ 8 $\begingroup$Let $E,f:E\to\mathbb{R},L,c$ be things. If for all $\varepsilon>0$, there exists $\delta>0$, such that for all $x\in E$, $0<|x-c|<\delta\rightarrow |f(x)-L|<\varepsilon$", then we say $f(x)$ converges to $L$ when $x$ approaches $c$.
Some of this is a matter of style - which isn't very important. But there are a few "technical" problems with some of your examples.
Examples 2.1 and 2.2:
If for all $x \in A$, $x \in B$, then ...
If you parse this as "If ( for all $x \in A$ and $x \in B$ ) ..." it doesn't make sense. The intended meaning is "For all $x$, if $x \in A$ implies that $x \in B$, we say ...".
Since you are considering the $x$'s "one at a time" here, I would prefer "for each/any/every $x$" to "for all $x$".
Example 3.2:
I would prefer "for each/any/every subspace" to "for all subspaces", for the same reason as above.
Examples 4.1, 4.2
The grammar seems a bit convoluted here. The main statement is of the form "If (the premises) then (the conclusion)" but you have a subsidiary clause (stating the properties of $X$ and $Y$) after the premises. I would re-order 4.1 as something like
If $X$ is compact and $Y$ is Hausdorff, then every bijective continuous function $f:X \to Y$ is a homeomorphism.
and similarly for 4.2.
Example 5.3
"If let $\epsilon > 0$ ..." is not correct English grammar. You need a clause after the "If", but "let $\epsilon > 0$ ... then $|f(x)−L|< \epsilon$" is a complete sentence, not a clause within the bigger sentence starting with "If".
$\endgroup$ 4 $\begingroup$One instance where this can matter is when working with formal logic, particularly quantifier logic. In that case, $\forall$ has a very specific meaning, and it is very important to clearly distinguish the formal logic itself and the meta-logic of proving things about the formal logic. For that reason, it may be clearer to avoid the phrase "for all" entirely from the meta-logic, to avoid any confusion with the $\forall$ symbol.
$\endgroup$ 2 $\begingroup$I might misunderstand your question, and I'm not a logician, but I can't resist to give an answer here. In my point of view it is more important to be clear and easy to read than to write a logically 100% correct statement.
I think that in some of your examples, you should use neither "Let" or "For all", but rather use words, to make the statements easier to digest (I'm well aware, and respect that others think different). Suggestions:
Example 1
The square of an even number is even.
If it is not clear enough that this holds for all even numbers, then maybe:
The square of every even number is again even.
Example 2
We say that $A$ is a subset of $B$ if every element of $A$ also belongs to $B$.
Example 4
$\endgroup$ 2Every bijective continuous mapping from a compact space to a Hausdorff space is a homeomorphism.
