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I would assume the answer to my question is yes, but I want to make sure because my book uses both terminologies. Please also indicate where zero falls into the mix. UPDATE: Here is an excerpt from my book:
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The real numbers can be partitioned into the positive real numbers, the negative real numbers, and zero. A real number is one and only one of those three possibilities. This is called "trichotomy." Non-negative (or, correspondingly, non-positive) means not negative (not positive), so zero or positive (zero or negative). That is, non-negative includes zero whereas positive does not. |
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In mathematical English,
So non-negative means $\ge 0$, not the same as positive. In mathematical French, it just happens that the word 'positif' is defined to be $\ge 0$, that is, 0 is both 'positif' and 'negatif'. In other languages...who knows. |
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If we go by your edits, about the book excerpt, it looks like the book treats non-negative as $\ge 0$, and positive as $\gt 0$. Also, from the notation it seems like you are talking about functions whose domain is $\mathbb{N}$. For an example of an asymptotically positive function, consider $$ f(n) = 1$$ For an example of an asymptotically non-negative function, consider $$f(n) = \left|\sin\left(\frac{n\pi}{2}\right)\right|$$ For sufficiently large $\displaystyle n$, we have that $\displaystyle f(n) \ge 0$. Note that this function is not asymptotically positive, because it is zero (for even $\displaystyle n$) infinitely often. Any asymptotically positive function is also asymptotically non-negative, but not vice-versa. For an example of a function which is neither asymptotically non-negative, nor asymptotically positive, $$f(n) = \sin\left(\frac{n\pi}{2}\right)$$ This function takes the values $\displaystyle 1,-1 \ \text{and}\ 0$ infinitely often. |
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As non-negative is an adjective, generally its meaning depends on what word comes after it. |
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