TITLE: Is the set of natural numbers closed under subtraction?
QUESTION [8 upvotes]: I was reading my textbook right now on elementary number theory, and this question came up. It got me wondering: How can subtraction be performed on two elements $x,y\in\mathbb{N}$ if subtraction is defined by $x-y=x+(-y)$, and $\forall a\in\mathbb{N}, 0<a<\infty$?
I realize that $1-10=-9$, which is not in $\mathbb{N}$. But the question just bothered me, because I wonder if you can technically subtract two elements of $\mathbb{N}$?

REPLY [2 votes]: You can prove that for any natural numbers $a,b$ there is at most one $c$ such that $b+c=a$. That means that we can define $z=a-b$ to mean $z+b=a$. This means that if $z=a-b$ and $w=a-b$ then $z=w$ - that is, subtraction is well-defined when it is defined.
You could also just define the integers based on the naturals, and define subtraction there for all pairs of integers.