Operations of the natural numbers

We're going to see the operations of the natural

With the axioms of Peano and with the zero as the first element of \mathbb{N}_0, we introduce the notation of others:

\begin{cases}s(0)\text{ we call 1} \\ s(1)\text{ we call 2} \\ \cdots \\ s(n-1)\text{ we call n}\end{cases}

Operation sum

The sum or (addition) in \mathbb{N}_0 it is an operation that +|\mathbb{N}_0\rightarrow \mathbb{N}_0 which is recursively defined as:

\begin{cases}a+0=a \\ a+s(b)=s(a+b) \end{cases}

Properties of the sum

Dice a, b, c\in\mathbb{N}_0 comply with:

• Associative property for addition
  a+(b+c)-(a+b)+c (as a result of the previous property, there is no need to enter parentheses and a+b+c can be written)
• Commutative property for addition
  a+b=b+a
• Neutral element of the sum
  0\in\mathbb{N}_0,\forall a\in\mathbb{N}_0|a+0=a; \forall a\in\mathbb{N}_0, \exists b\in\mathbb{N}_0|a+b=0
• Cancellation property (simplification) in the sum
  \text{If }a+c=b+c\rightarrow a=b

Operation product

The product or (multiplication) in \mathbb{N}_0 it is an operation that \cdot|\mathbb{N}_0\rightarrow \mathbb{N}_0 which is recursively defined as:

\begin{cases}a\cdot 0=0 \\ a\cdot s(b)=a+a\cdot b \end{cases}

Properties of the product

Dice a, b, c\in\mathbb{N}_0 comply with:

• Associative property for the product
  a\cdot(b\cdot c)=(a\cdot b)\cdot c (as a result of the above property, there is no need to indicate parentheses and can be written a\cdot b\cdot c)
• Commutative property for the product
  a\cdot b=b\cdot a
• Neutral element of the product
  1\in\mathbb{N}_0,\forall a\in\mathbb{N}_0|a\cdot 1=a; \forall a\in\mathbb{N}_0,a\not=0, \exists b\in\mathbb{N}_0|a\cdot b=1
• Cancellation property (simplification) on the product
  \text{If }a\cdot c=b\cdot c\text{ con }c\not= 0\rightarrow a=b

Operation sum and product

In addition, sum and product share the following property:

Property of the sum and the product

• Distributive property of the product regarding the sum
  \text{If }a\cdot (b+c)=(a\cdot b)+(a\cdot c)

Note if we had chosen to represent the natural ones without zero, in the axioms of Peano it would be enough to change \mathbb{N}_0 by \mathbb{N} and 0 by 1. Since the first axiom only serves to ensure that \mathbb{N}_0 is not the empty set, and the name given to the first element is not relevant in its definition. When it becomes relevant, it is when the sum and product operations are defined. If the natural ones are built starting at 1, the definition of the sum is started with a+1 s(a), and that of the product by a\cdot 1=a