Anita Tabacco include zero to N?

Question

I have two problems actually, first one: Proffesor always grumbled that Anita Tabacco include zero to natural numbers, and proffesor dont include zero to the natural numbers when explain somethink or give any examples, I wasnt at his first lecture, so i dont know the reason of its statement by proffesor

Another one: Today while we study absolute value there were notion or i dont know how its called:

and me dear proffesor said that the case when $x>or=0$ is a wrong and $x$ only more than zero, after give some examples by using $sin$ but trigonometry is my achilles' heel and i just couldnt keep up after his thoughts.

If someone have any idea about it please, I in need of help.

On the "thorny" issue about $0 \in \mathbb N$ or not, see this post, — Mauro ALLEGRANZA, Oct 02 '15 at 14:57
Related : Claudio Canuto & Anita Tabacco, Mathematical Analysis I (2015). — Mauro ALLEGRANZA, Oct 02 '15 at 15:01

score 4 · Accepted Answer · answered Oct 02 '15 at 14:56

You can define the absolute value as $$ |x| = \begin{cases} x &\text{if }x\ge 0 \\ -x &\text{if }x<0 \end{cases} $$ or $$ |x| = \begin{cases} x &\text{if }x > 0 \\ 0 & \text{if }x=0 \\ -x &\text{if }x<0 \end{cases} $$ or even $$ |x| = \begin{cases} x &\text{if }x > 0 \\ -x &\text{if }x\le 0\end{cases} $$

Since they all define exactly the same function (they all make $|0|=0$, just with different words), they are all equally correct.

Anita Tabacco include zero to N?

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