Syntax for negation of a mathematical statement

Question

My question is fairly simple. I aim to find the negation of a statement that reads:

$$\forall\epsilon\in\mathbb{R}, \exists\delta\in\mathbb{R} : (|x - \pi| < \delta) \implies (|\sin(x)| < \epsilon)$$

Would the negation of this statement take the form

$$\exists\epsilon, \delta\in\mathbb{R} : (|x - \pi| < \delta) \wedge (|\sin(x)| \geq \epsilon)$$

or

$$\exists\epsilon\in\mathbb{R} : \forall \delta \in \mathbb{R}, (|x - \pi| < \delta) \wedge (|\sin(x)| \geq \epsilon) ?$$

I'm specifically wondering whether or not the negation of a "for all x, there exists y such that P" would equate to "there exists an x and y such that (not P)", or "there exists an x, where for all y, (not P)".

Apologies if this is rather obvious or a "yes or no" question, I generally have difficulty reading all the notation smushed together, so I'm a bit lost as to what would be the logical negation to the statement.

Answer 1

I generally have difficulty reading all the notation smushed together

Fair enough - so "unsmush" it. I mean, take it one bit at a time.

• The negation of $\forall x\in S\ \langle\hbox{whatever}\rangle$ is $\exists x\in S\ \langle\hbox{not (whatever)}\rangle$.

• The negation of $\exists x\in S\ \langle\hbox{whatever}\rangle$ is $\forall x\in S\ \langle\hbox{not (whatever)}\rangle$.

• The negation of $p\Rightarrow q$ is $p\wedge(\hbox{not}\ q)$.

Doing these one at a time, you get the negation of your statement as $$\exists\epsilon\in{\Bbb R}\ \hbox{not}\ \Bigl[ \exists\delta\in{\Bbb R}\ (|x-\pi|<\delta)\Rightarrow(|\sin x|<\epsilon)\Bigr]$$ and then $$\exists\epsilon\in{\Bbb R}\ \forall\delta\in{\Bbb R}\ \hbox{not}\ \Bigl[(|x-\pi|<\delta)\Rightarrow(|\sin x|<\epsilon)\Bigr]$$ and finally $$\exists\epsilon\in{\Bbb R}\ \forall\delta\in{\Bbb R}\ (|x-\pi|<\delta)\wedge(|\sin x|\ge\epsilon)\ .$$

Answer 2

Imagine an analogous statement in ordinary English, for example

Every boy has a brother.

That happens to be false. The way to negate it is to show

There is a boy who has no brother.

An uglier way to write these statements is to make them match the formal hard to parse statements you see in some mathematics ("all the notation smushed together"):

For every boy there is another boy who is that boy's brother

to which the ugly negation is

There is some boy such that no other boy is his brother.

Can you use this logical template to answer your question?

Answer 3

$$\forall\epsilon\in\mathbb{R}, \exists\delta\in\mathbb{R} : (|x - \pi| < \delta) \implies (|\sin(x)| < \epsilon)$$ Would its negation take the form $$\exists\epsilon\in\mathbb{R} : \forall \delta \in \mathbb{R}, (|x - \pi| < \delta) \wedge (|\sin(x)| \geq \epsilon) ?$$

To add to what has been said, the original statement really ought to be

$$\forall\epsilon{\in}\mathbb{R}\; \exists\delta{\in}\mathbb{R} \;\color\red{\forall x{\in}\mathbb{R}}\; \big(|x - \pi| < \delta \implies |\sin(x)| < \epsilon\big),$$

and its negation thus $$\exists\epsilon{\in}\mathbb{R}\; \forall\delta{\in}\mathbb{R} \;\color\red{\exists x{\in}\mathbb{R}}\; \big(|x - \pi| < \delta \;\land\; |\sin(x)| \ge \epsilon\big).$$

Besides the crucially missing quantification of $x,$ I recommend dropping that comma and colon, because treating formal logic as a literal shorthand for natural language (importing its punctuation and hanging quantifiers) can lead to ambiguities; note that this point doesn't contradict Ethan's helpful advice.

the question is posed exactly as written in the textbook

If that conditional has been introduced with the word "whenever", or set off (block-formatted away) from the rest of its sentence, then the $\forall x$ can be understood to be implicit.