AXIOMS OF REAL NUMBERS : REAL ANALYSIS #axioms #of #real #numbers
Axioms of real numbers:
The structure of real analysis rests on the real number system. It is a development of the set of real numbers starting from the set of natural numbers.
We consider real numbers as undefined elements satisfying
I. Field axioms
II. Order axioms
III. Completeness axioms
Field axioms:
A field is a set F with two operations ,called addition & multiplication, which satisfy the following so-called "field axioms".
Axioms of addition:
* If xꜪF, yꜪF implies x+y Ꜫ F
* Addition is commutative: x+y=y+x for every x,y in F.
* Addition is associative: (x+y)+z=x+(y+z) for every x,y,z in F.
* F contains an element O such that O+x=x+O for every x in F.
here O is called the additive identity in F.
* For every x Ꜫ F , there exist -x Ꜫ F such that x+(-x)=0.
here '-x' is called the additive inverse of x in F.
Axioms of multiplication :
* if x,y Ꜫ F implies that x.y Ꜫ F.
* multiplication is associative : (x.y).z=x.(y.z) for every x,y,z in F.
* multiplication is commutative : x.y=y.x for every x,y in F.
* F contains an element 1 ≠ 0 such that 1.x=x for every x in F.
Here 1 is called the multiplicative identity in F.
* If x ≠ 0 in F, there exists and element 1/x in F such that x.(1/x) =1.
Here 1/x is called the multiplicative inverse of x in F.
The distributive properties :
for x,y,z Ꜫ F, x.(y+z) = x.y + x. z ( right distribution )
and (x+y) .z = x.z + y.z ( left distribution )
Examples and Counter Examples :
* The set N of all natural numbers is not field because the additive identity 1 is not in F.
* The set Z of all integers is not a field because for any x in F, the multiplicative inverse1/x is not in Z.
* The field of axioms clearly hold in the set Q of all rational numbers and the set R of all real numbers.
Order Axioms :
The term "ORDER " referred as a relation between two numbers such as > , < , = ,etc i.e. for any two real numbers x,y we have x<y or x=y or y<x. Here the symbol < is referred as an order relation.
If we take one relation > as order relation, it has the following properties :
* For a,b Ꜫ R , a>b or b>a or a=b → Law of Trichotomy
* For a, b Ꜫ R, a > b , b > c implies a > c → Transitivity
* For a, b ,c Ꜫ R , if a > b then a+c > b+c → Monotone property for addition
also i) a > b implies ac > bc when c > 0
ii) a > b implies ac < bc when c <0
The above two conditions i and ii are called as monotone properties for multiplication.
Since the order relation exists in R, R is called a linearly ordered field.
Completeness Axioms :
The completeness axiom is stated as " Every non-empty set of Real numbers which is bounded above has a supremum ".
Since the set R of all real numbers satisfies the field , order and completeness axioms hence the set R is called a complete ordered field.
#axioms #of #real #numbers
#field #order # completeness #supremum #addition #multiplication
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