Table of contents |
2 The Properties |
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.
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
The Definition
The Axioms
The first is referred as AP1, the second as AP2.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.
Base: a+0=a=0+a and a+1=a+=1+a for all a
Proof of base is by mathematical induction on a.
Induction hypothesis: a+b=b+a for all a