# Proving that the set of positive integers does not form a group under addition

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.

• What is your definition of a group? (In particular, the part relating to existence of an inverse) – Clement C. Jun 5 '16 at 16:47