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 $\displaystyle \varepsilon > 0$ there exists $\displaystyle \delta > 0$ such that

whenever x and t are in D and satisfy $\displaystyle \lvert x-t \rvert < \delta $

, then $\displaystyle \lvert f(x) - f(t) \rvert < \varepsilon $

3)For all $\displaystyle \varepsilon > 0 $ there exists $\displaystyle \delta > 0$ such that

whenever $\displaystyle x \in D $ and $\displaystyle 0 <\lvert x-a \rvert < \delta $,

then $\displaystyle \lvert f(x)-A \rvert < \varepsilon $

Following are my negated statements

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

2)There exists $\displaystyle \varepsilon > 0 $ such that for every

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

3)There exists $\displaystyle \varepsilon > 0 $ such that for every

$\displaystyle \delta >0$ there exists $\displaystyle x\in D$ such that

$\displaystyle 0 <\lvert x-a \rvert < \delta $ and $\displaystyle \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