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.
- 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
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^{+})
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^{+})
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
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