Axioms of Real Number System

Last Updated : 26 Apr, 2021

In this article, we shall look at some very basic ideas about the Real Analysis, i.e. the study of the structure of Real Number System. We shall discuss the three axioms that are considered to be satisfied by the set of Real Numbers, $R$

The three axioms are :

Field Axioms
Order Axioms
Completeness Axiom

Field Axioms : The set $R$ is represented as a field $(R, +, .)$ where $+$ and $.$ are the binary operations of addition and multiplication respectively. It consists of 4 axioms for addition and multiplication each and one distributive law.

(i) Axioms for addition :

$a+b = b+a \ ∀ \ a,b \ ∈ R$
$(a+b)+c = a+(b+c) \ ∀ \ a,b,c \ ∈ R$
R contains an element 0 such that $a + 0 = a \ ∀ \ a ∈ R$
For each $a ∈ R$ there corresponds an element $-a ∈ R$ such that $a+(-a) = 0$

(ii) Axioms for multiplication :

$ab = ba \ ∀\ a,b ∈ R$
$(ab)c = a(bc) \ ∀ \ a,b,c ∈ R$
$R$ contains an element $1$ such that $1.a = a \ ∀ \ a ∈ R$ and $1 ≠ 0$
If $a ∈ R \ and \ a≠0$ then there exists an element $\frac{1}{a} ∈ R$ such that $a. (\frac{1}{a} ) = 1$

(iii) The distributive law :

$a(b+c) = ab+ac \ ∀ \ a,b,c ∈ R$

Order Axioms : We define $>$ (Greater Than) as the order relation, and it satisfies the following axioms –

Law of Trichotomy – For $a,b∈ R$ only one of the expressions can be true : $a>b , a=b , b>a$
Transitivity – For $a,b,c∈R \ a>b,\ b>c ⇒ a>c$
Monotone Property for addition – For $a,b,c∈R, \ a>b ⇒ a+c > b+c$
Monotone Property for multiplication – For $a,b,c∈R,\ a>b, c>0 ⇒ ac > bc$

We call $>$ linear order and $R$ is called a linearly ordered field.

Before defining the Completeness Axiom, we shall look at the concept of Boundedness. Here, we shall define a few terms before stating the Completeness Axiom.

Aggregate : Any non empty subset, say $A$ , of $R$ is known as an aggregate. For example, the set $Z^+$ is an aggregate. Similarly, the set B = {1,2,4,8} is also an aggregate since $B ⊆ R$ But, the set A = {x,y,z} and the empty set $∅$ are not aggregates.

Upper bound : A subset $S$ of $R$ is said to be bounded above if $∃ \ k_1\ ∈ R$ such that $x ∈ S ⇒ x \leq k_1$ . This number $k_1$ is called an upper bound of $S$ . For example, the set $R^-$ of negative real numbers is bounded above and $0$ is an upper bound. Similarly, the set $Z^-$ of negative integers is bounded above and $-1$ is the upper bound. But, the set $R^+$ of positive real numbers is not bounded above.

Lower Bound : A subset $S$ of $R$ is said to be bounded below if $∃ \ k_2\ ∈ R$ such that $x ∈ S ⇒ x \geq k_2$ This number $k_2$ is called a lower bound of S. For example, the set $R^+$ is bounded below and $0$ is a lower bound. Similarly, the set $Z^+$ is bounded below and $1$ is the upper bound. But, the set $R^-$ is not bounded below.

Least Upper Bound : Consider an upper bound $u$ of an aggregate $S$ and any real number less than $u$ is not an upper bound of $S$ , then we say $u$ is the least upper bound(lub) or supremum(sup) of $S.$

Greatest Lower bound : Consider a lower bound $v$ of an aggregate $S$ and any real number greater than $v$ is not a lower bound of $S$ , then we say $v$ is the greatest lower bound(glb) or infimum(inf) of $S.$

Example : Let $S = [0,1]$ . For S, we see that 1 is an upper bound and any number less than 1 is not an upper bound of S, hence, 1 is supremum of S. Also, 0 is a lower bound and any number greater than 0 is not a lower bound, so, 0 is infimum of S.

Boundedness : An aggregate S is bounded if it is both bounded above and bounded below. That is, it must have both an upper bound and a lower bound. For example, any finite set is bounded, the empty set $∅$ is bounded. But, the sets $Q$ and $R$ are not bounded.

Note : An aggregate need not have a greatest and a least member to be bounded above or bounded below respectively.

Now being done with the required definition, we state the Completeness Axiom(also called the least upper bound axiom).

” Every non-empty set of Real numbers which is bounded above has a supremum.”

The set R satisfies the Field Axioms, Order Axioms, and the Completeness Axiom. Hence the set of real numbers $R$ is called a complete ordered field.

Also, the set of rational numbers, $Q$ does not satisfy the completeness axiom. Hence, $Q$ is not a complete field.

The completeness axiom is a really fundamental and important property of real number systems, as proofs various theorems of calculus, the concepts of maxima and minima, mean-value theorems etc. rely on the completeness property of real numbers.

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