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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:

The definition of $\Theta(g(n))$ requires that every member $f(n) \in \Theta(g(n))$ be asymptotically non-negative, that is, that $f(n)$ be non-negative whenever n is sufficiently large. (An asymptotically positive function is one that is positive for all sufficiently large $n$.)

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    @Willie — I was meaning the *open* sense is including 0. As opposed to *strict*. But I got it ! You talk in the topological sense of *open set*. ;-) Your friends and you are more mathy than me.2014-05-05

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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.

Edit for clarity:

Non-negative means zero or positive.

Non-positive means zero or negative.

That is, non-negative includes zero whereas positive does not and vice versa.

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    @ubiquibacon — You call that clarifying ? ;-)2014-05-04
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In mathematical English,

  • positive is defined to be $> 0$
  • negative is defined to be $< 0$

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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    @muntoo: :) part of the issue is that naming conventions are just that, conventions, and not necessarily logical. The mathematical concepts are, but the labels not necessarily so; for the concept of trichotomy English goes one way, French another, other languages may have another convention. Some countries drive on ne side of the road, some on the other, and it all works out as long as you know the rules of the country you're in.2011-01-22
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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.