Answer to Question #125040 in Algebra for Sphesihle Zuma

Question

Discuss the steps that you can deploy in constructing number problems.

Answer 1

$\to$ Construction of number system

If we want fully understand numbers and number sets, we must learn how to construct each sets of numbers.

The set of natural numbers is axiomatic introduced by Italian mathematician Giuseppe Peano. These axioms are known as Peano’s axioms.

Construction of number systems: Natural numbersPeano Axioms

N  is a set with the following properties:

1.  N has a distinguished element which we call (1).
2. There exists  s: N$\to$ N
3. s is one-to-one function (injective function)
4. There does not exists an element x$\in$ N such that s(x)=1 (nor surjective function)
5. is M $\subset$ N such that : ( priciple of induction )

(a). 1 $\isin$ S,

(b). if x $\isin$ M $\implies$ s(x) $\in$ S ,

the set M=N .

A set N is the set of natural numbers and its element are natural numbers. A function s

 is called a successor function.