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 Definition

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 Axioms

• a+0 = a
• a+(b⁺) = (a+b)⁺

The first is referred as AP1, the second as AP2.

The Properties

• Uniqueness: (a+b) is unique. i.e. If (a.b) also satisfies [AP1] and [AP2] then (a.b)=(a+b).
• The Law of Associativity: (a+b)+c = a+(b+c)
• The Law of Commutativity: a+b = b+a

Proof of Uniqueness

We prove by mathematical induction on b.

Base: (a.0) = [by AP1] a = [by AP1] (a+0) for all a

Induction hypothese: (a.b)=(a+b) for all a

(a.b⁺)

= [by AP2] (a.b)⁺

= [by hypothese] (a+b)⁺

= [by AP2] (a+b⁺)

Proof of Associativity

We prove by mathematical induction on c.

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

(a+b)+c⁺

= [by AP2] ((a+b)+c)⁺

= [by hypothesis] (a+(b+c))⁺

= [by AP2] a+(b+c)⁺

= [by AP2] a+(b+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

a+b⁺

= [using the base] a+(1+b)

= [by associativity] (a+1)+b

= [by hypothesis] b+(a+1)

= [using the base] b+(1+a)

= [by associativity] (b+1)+a

= [using the base] b⁺+a