# Math Help - Need to check my work

1. ## Need to check my work

Hi

I want to negate the following statements. Please check if I am doing it correctly.

1)There exists p > 0 such that for every x we have f(x+p)= f(x)

2) For all $\varepsilon > 0$ there exists $\delta > 0$ such that
whenever x and t are in D and satisfy $\lvert x-t \rvert < \delta$
, then $\lvert f(x) - f(t) \rvert < \varepsilon$

3)For all $\varepsilon > 0$ there exists $\delta > 0$ such that
whenever $x \in D$ and $0 <\lvert x-a \rvert < \delta$,
then $\lvert f(x)-A \rvert < \varepsilon$

Following are my negated statements

1)For all $p> 0$ there exists x such that $f(x+p)\neq f(x)$

2)There exists $\varepsilon > 0$ such that for every
$\delta >0$ there exists x and t in D such that $\lvert x-t \rvert < \delta$ and $\lvert f(x) - f(t) \rvert \geqslant \varepsilon$

3)There exists $\varepsilon > 0$ such that for every
$\delta >0$ there exists $x\in D$ such that
$0 <\lvert x-a \rvert < \delta$ and $\lvert f(x)-A \rvert \geqslant \varepsilon$

Please tell me if I am doing it right. I have followed the examples given in my book.
The book is "A Friendly Introduction to Analysis: Single and Multivariable, second edition" Author- Witold Kosmala
I am not satisfied with his treatment of the topic there. He discusses these things under the topic of "Proof Techniques", where he also talk talks about methods of proofs, like contradiction , contrapositive.

If you can suggest a good source on internet ( preferably free) where they discuss
negation of the mathematical statements, please suggest me.

thanks

newton

2. Yes, you are correct.

Negating such statements is pretty easy. They usually have the form ∀x (P(x) → Q(x)) or ∃x (P(x) ∧ Q(x)) where Q(x) either is an atomic formula (equality, inequality, etc.) or in turn has the same form. We have the following equivalences.

¬(A → B) ⇔ A ∧ ¬B
¬(A ∧ B) ⇔ A → ¬B
¬∀x A(x) ⇔ ∃x ¬A(x)
¬∃x A(x) ⇔ ∀x ¬A(x)

Therefore,

¬(∀x (P(x) → Q(x))) ⇔ ∃x (P(x) ∧ ¬Q(x))
¬(∃x (P(x) ∧ Q(x))) ⇔ ∀x (P(x) → ¬Q(x))

Thus, to negate a proposition that starts with a sequence of quantifiers, you switch the quantifiers without changing the restrictions on quantified variables (P(x) above, e.g., x ∈ D, δ > 0 or |x - t| < δ) and negate only the last atomic formula.

3. Priviet Makarov

Thanks. That makes sense.... I never had a logic course. Can you suggest some good book on logic for self study ?

newton

4. Originally Posted by issacnewton
Priviet Makarov

Thanks. That makes sense.... I never had a logic course. Can you suggest some good book on logic for self study ?

newton

Introduction to Logic by Irving-M-Copi.

5. thanks zarathustra