The reason why m+n is defined for all m, n in the set of natural numbers is because the set of natural numbers is infinite. Any length m added to another length, n, will always exist.
The binary operation +:NxN→N is commutative. We can prove this geometrically because if we first have a length m and add it to another length n, and in another situation we first have a length n and add it to another length m, the 2 lengths, m+n and n+m will be the same. I show this below.
The binary operation +:NxN→N is associative. I prove this by using a similar method that I will also show below.
The cancellation property for all m, n, p in the set of Natural numbers means that m + p = n + p means that m = n. I will prove this by using the fact that a + 0 = a for all a in the set of natural numbers. I will also use the associative property to prove this.
The Grothendieck group of (N, +) exists because
The reason why m x n exists for all m, n in the set of natural numbers is because the set of natural numbers is infinite and m groups of n or n groups of m will exist for all values of m and n.
The binary operation x:NxN→N is commutative. The reason that m x n = n x m is because conceptually, m groups of n and n groups of m give the same value.
The binary operation x:NxN→N is associative. The reason that (m x n) x p = m x (n x p)
The reason that m x 1 = m = 1 x m is because if we think of m x 1 and 1 x m as an array of m objects, whether the array is going vertical with m objects and only one column of the objects or the array is going horizontal with m objects and only one row, the array will only have m objects.
The reason why the cancellation property of multiplication exists, that is if m x p = n x p, then m = n for all m, n, p in the set of natural numbers, is because if we can divide both sides of the equation by p. In other words, we can multiply both sides of the equation by 1/p which is the reciprocal of p and this is the same thing as dividing by p. m x p x 1/p = m x 1 because p x 1/p = p/p and any number divided by itself is 1. Now m x 1 = m. This is the left hand side of the equation. n x p x 1/p = n x 1 because p x 1/p = p/p and any number divided by itself is 1. Now n x 1 = n. This is the right hand side of the equation. Now m = n.
a/b is logically equivalent to a/b because if we multiply both sides of the equality by 1/b, we will get a/b x 1/b is logically equivalent to a/b x 1/b. This simplifies to a x 1 = a x 1 because b x 1/b = b/b = 1 since any number divided by itself is 1. Then a x 1 = a x 1 simplifies to a = a because any number multiplied by 1 equals itself. Also from what is said before in the previous lines of this mathematica notebook, a/b is logically equivalent c/d means a x d = b x c. In this case, because a/b is logically equivalent to a/b, this means that a x b = a x b which is true.
If a/b is logically equivalent to c/d, then this means that a x d = b x c. If c/d is logically equivalent to a/b, then this means that b x c = a x d. Because multiplication is commutative, that means that a x d = b x c and b x c = a x d. This means that if a/b is logically equivalent to c/d then if c/d is logically equivalent to a/b.
If a/b is logically equivalent to c/d and c/d is logically equivalent to e/f, then a/b is logically equivalent to e/f. This is true because a x d = b x c and c x f = e x d. Then c = d x e / f if we divide both sides of the second equation by f. Then if we substitute d x e /f in for c in a x d = b x c, we get a x d = b x d x e / f. I have written out the rest of the problem by hand so that it is more clear what is happening and how I did the problem.