Naive Question About Quantifiers: for all, for some, where, and given...

Question

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..

Answer 1

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.

Answer 2

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.