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I was bumped into a question related to quantifiers: and was wondering if anyone can give me a further explanation for the following four statements:

Let $f: \mathbb{R} \to \mathbb{R}$ be a function,

Statement 1: Given $\epsilon>0$, $|f(x)|<\varepsilon$

Statment 2: $|f(x)|<\varepsilon$ for all $\varepsilon>0$

Statment 3: $|f(x)|<\varepsilon$ where $\varepsilon >0$.

Statment 4: $|f(x)| < \varepsilon$ for some $\varepsilon>0$.

I think statement 1 and statement 2 are equivalent and statement 3 and stamens 4 are equivalent. Is my understanding correct?

Thank you.

[Updated Comment] For sure this is not a homework question, since I been argued with my friend about this topic so I'm trying to seek a more crystal clear way to enhance my understanding..

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Fianra
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    Statements 1 and 3 are just bad mathematical English. In 1, it is just about conceivable to read "Given" as a universal quantifier, but in 3 reading "where" as an existential or universal quantifier is impossible. – Rob Arthan Dec 29 '15 at 23:26
  • @RobArthan , Thanks for reply. I agree statement 1 and 3 are bad mathematical english. But I try to understand the meaning behind the "word": when a statement using "where" as my question stated, should I treat it as there exists $\epsilon>0$ such that $f(x)|<\varepsilon$? – Fianra Dec 29 '15 at 23:37
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    I think 1. is ambiguous by itself. If this is the first mention of $\varepsilon$ it means 2. but it's not great mathematical English. 3. is worse mathematical English, and is even more ambiguous — it might mean 2., and it might mean 4., though without more context I'd say it probably means 2. – BrianO Dec 30 '15 at 09:20
  • @RobArthan Here, I talked about the ambiguity of "where", showing an example of (even) a (good) mathematics teacher incoherently conflating it with both "for each" and "for some" within the same sentence. Moral of the story: in technical writing, unless one (has strong command of the English language and) is certain that no ambiguity ensues, best to avoid using 'where' as a qualifier. – ryang Jul 20 '22 at 05:45

2 Answers2

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Statements 2 and 4 are clear, and not equivalent. The other two hurt my head, as a mathematician. You would not be very likely to encounter those particular phrases in idiomatic mathematical English. Here is why.

Given

The word "given" is used to prove a statement involving a universal quantifier. For example:

Theorem: for every even natural number $a$, $3a$ is also even

Proof. Given an even natural number $a$, there is a natural number $n$ such that $a = 2n$. Then $3a = 3(2n) = 2(3n)$. So $3a$ is also even. $\Box$

The use of "given" here shows that the proof is intended to work for every object of some kind, thus proving the universal statement.

If I had to read statement 1 as a standalone statement, I would read the "given" as a universal quantifier. But that is probably not what should be meant, because if statement 1 was true in that reading then we would have $|f(x)|= 0$.

Where

The word "where" is often used to specify a property of an object that has just been chosen. Statement 3 is not idiomatic to me, because it seems to me that it is trying to choose $\epsilon$ given $x$. I would need more context to know what is intended.

For example, we would use "where" to state a property of the derivative:

If $f'(a) = m$ then we have $f(x + a) = f(a) + m(x-a) + h(x)$ where $h$ is a function such that $\lim_{x \to a} h(x)/(x-a) = 0$.

In that property, the "where" means "for some". But statement 3 above just doesn't read well to me - I would ask a student to rephrase it, if they submitted it in homework.

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Carl Mummert
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No. Statements 1, 2, and 3 are equivalent. "Given" can be viewed as "Given any" a.k.a for all, (the statement which says for all should be obvious) and "where" is saying "in the cases where $\epsilon>0$" which is again the same as for all.

In this instance "for some" is not "for all," this is merely stating the existence of an $\epsilon$, not making a statement about all of them.

AnotherPerson
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    You are being far too kind to the rubbish mathematical English that the OP is being asked to consider (and you are being a little unkind to the OP to say "No". See my comments on the question). – Rob Arthan Dec 29 '15 at 23:27