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The set of positive integers under addition has closure because it goes on for infinity and you will always have the element $a + b$. I am also aware that addition is associative but do we include $0$ in this group? It is neither positive nor negative so I would have assumed not.

I've been given the answer by a textbook that there is "No inverse of $-1$". But I don't even see how that relates the set of positive integers. My answer would have been that there was no identity or that all the elements do not have an inverse.

Could someone clarify this please?

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    $\begingroup$ What is your definition of a group? (In particular, the part relating to existence of an inverse) $\endgroup$ – Clement C. Jun 5 '16 at 16:47
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Whether or not you include 0 as a positive integer is largely a matter of convention - I think most people would say "no".

If you don't include 0, then you are correct that the absence of an additive identity means it's not a group.

There are also no inverses, as you note, whether or not you include 0.

The answer given in the book should have read "There is no inverse for 1", not "There is no inverse for -1".

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    $\begingroup$ 0 is certainly never a positive integer. Whether it is a "natural number" is not as certain. $\endgroup$ – Max Jun 5 '16 at 16:53
  • $\begingroup$ Hm, I've definitely heard the phrases "strictly positive" and "non-negative" to distinguish between the two cases, which seems silly if people never consider 0 to be positive. However, I think 0 is almost never included when people say "positive integer", as you suggest. $\endgroup$ – Josh Hunt Jun 6 '16 at 15:29

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