Addition in N
Addition of natural numbers is the most basic arithmetic operation. Here we will define it
from Peano's axioms (see natural number) and prove some simple properties. The set of natural numbers will be denoted by N; zero
is taken to be a natural number.
The operation of addition, commonly written as infix operator +, is a
function of N x N -> N
a + b = c
a is called the augend, b is called the addend, while c is called the sum.
By convention, a+ is referred as the successor of a as defined
in the Peano postulates.
The first is referred as AP1, the second as AP2.
Base: (a.0) = [by AP1] a = [by AP1] (a+0) for all a
Induction hypothese: (a.b)=(a+b) for all a
Base: (a+b)+0 = [by AP1] a+b = [by AP1] a+(b+0) for all a,b
Induction hypothesis: (a+b)+c = a+(b+c) for all a,b
Base: a+0=a=0+a and a+1=a+=1+a for all a
Induction hypothesis: a+b=b+a for all a
Table of contents
1 The Definition
The Definition The Axioms
The Properties Proof of Uniqueness
We prove by mathematical induction on b.
Proof of Associativity
We prove by mathematical induction on c.
Proof of Commutativity
We prove by mathematical induction on b.
Proof of base is by mathematical induction on a.