Is location of $\forall$ $\epsilon$ > 0 important?

Question

Using epsilon-delta do I have to write $\forall$ $\epsilon$ > 0 there exists $\delta$ such that..... rather than there exists $\delta$ such that ..... $\forall$ $\epsilon$ > 0? Is the location of $\forall$ $\epsilon$ > 0 important? I got minus grade because I wrote $\forall$ $\epsilon$ at the end of the sentence.

Answer 1

There is a difference between $\forall x \exists y$ and $\exists y \forall x$, and in fact there is an implicit set of parentheses around such qualifiers that we drop when we think their existence is understood.

The statement $\forall x \in \mathbb{N}(\exists y \in \mathbb{N}(x < y))$ (writing out all the parentheses) means "for every natural number $x$, there exists some natural number $y$ such that $x$ is less than $y$", which is a true statement - give me any natural number $x$, and I'll give you a bigger natural number.

By comparison, if we swap them around, we get $\exists y \in \mathbb{N}(\forall x \in \mathbb{N}(x < y))$, which says "there exists a natural number $y$ such that for every natural number $x$, $x$ is less than $y$", or in other words "there is a natural number that is bigger than every natural number", which is clearly false.

If you put the quantifier at the end of your statement, like $\exists y \in \mathbb{N} x < y \forall x \in \mathbb{N}$, then you're using non-conventional notation, and especially in the absence of any parentheses it's not obvious which part of the statement it belongs to. Are you trying to say $\exists y \in \mathbb{N} ((x < y) \forall x \in \mathbb{N})$, or $(\exists y \in \mathbb{N} (x < y)) \forall x \in \mathbb{N}$? By keeping things in the conventional order - put conditionals at the front, in the order of decreasing scope - you eliminate that ambiguity.

Answer 2

Hint: Which of the following is true:

1. There exists a nonzero real $y$ such that $xy>0$, for each nonzero real $x$.
2. For each nonzero real $x$, there exists a nonzero real $y$ such that $xy>0$.

That aside, in formal syntax, the convention is that we write quantifiers before the expression it quantifies. That is just how we write quantifiers. Similarly to how you $-x$ rather than $x-$ when you are talking about the inverse of $x$. This allows us to make formal expressions easier to read (and write) and less ambiguous (as illustrated by your own example!).

The caveat here is that frequently, it is best to not use formal notation at all, as most of the time, just spelling things out works best in writing. Formal quantifiers work well mostly in blackboard situations, and in cases when a given statement or formula is hard to unambiguously express in plain English.