# 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.

#### 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