Can someone help me out here: I'm a bit rusty on my set builder notation... Is the following correct?
The set of natural numbers isn't a group because a group is a set partially defined by { ∀a∃b | a + b = 0 } - or in other words for every element a in the group there exists an element b in the group such that a + b = 0; or to put it yet another way, every element a has an element b that is its arithmetic inverse. As the set of natural numbers are only positive, you can't satisfy this condition.
A group is a set equipped with a binary operator that obeys a few rules. One of the rules is the existence of inverses. If the operator is addition, then your description is correct. That is, every element x of the underlying set has an additive inverse x^-1 also in the set such that x + x^-1 = 0. Obviously, zero needs to be in the set as well and in some constructions of the natural numbers it is not.
The set of natural numbers isn't a group because a group is a set partially defined by { ∀a∃b | a + b = 0 } - or in other words for every element a in the group there exists an element b in the group such that a + b = 0; or to put it yet another way, every element a has an element b that is its arithmetic inverse. As the set of natural numbers are only positive, you can't satisfy this condition.