Construction of The Real Numbers

DEFINITION1: The set  \mathbb{R} with at least two distinct elements and satisfying the following five axioms is called the set of real numbers and each element of  \mathbb{R} is called a real number:

I. AXIOMS OF ADDITION:

The function  +:\mathbb{R}\times \mathbb{R}\to \mathbb{R} defined as  \left( x,y \right)\to x+y\in \mathbb{R} for each  \left( x,y \right) in  \mathbb{R}\times \mathbb{R} satisfies the following properties:

I {{}_{1}}.\,\forall a,b\in \mathbb{R},a+b=b+a,

I {{}_{2}}.\,\forall a,b,c\in \mathbb{R},a+(b+c)=(a+b)+c,

I {{}_{3}}.\,\exists 0\in \mathbb{R}:\forall a\in \mathbb{R},a+0=a ( 0 is called the additive identity element),

I {{}_{4}}.\,\forall a\in \mathbb{R},\exists b\in \mathbb{R}:a+b=0 ( b is called the additive inverse of  a).

A pair of  \left( X,+ \right) satisfying the properties  {{I}_{1}},{{I}_{2}},{{I}_{3}},{{I}_{4}} is called a commutative additive group (or called an Abelian group). According to this,  \mathbb{R} is a commutative additive group.

II. AXIOMS OF MULTIPLICATION:

The function  \cdot:\mathbb{R}\times \mathbb{R}\to \mathbb{R} defined as  \left( x,y \right)\to x\cdot{y}\in \mathbb{R} for each  \left( x,y \right) in  \mathbb{R}\times \mathbb{R} satisfies the following properties:

II {{}_{1}}.\,\forall a,b\in \mathbb{R},a\cdot b=b\cdot a,

II {{}_{2}}.\,\forall a,b,c\in \mathbb{R},a\cdot (b\cdot c)=(a\cdot b)\cdot c,

II {{}_{3}}.\,\exists 1\in \mathbb{R}:\forall a\in \mathbb{R},a\cdot 1=a ( 1 is called the multiplicative identity element),

II {{}_{4}}.\,\forall a\in \mathbb{R}\backslash \left\{ 0 \right\},\exists b\in \mathbb{R},a\cdot b=1 ( b is called the multiplicative inverse of  a).

The multiplication of  a and  b is generally denoted by  ab instead of  a\cdot{b}.

III. DISTRIBUTIVITY OF MULTIPLICATION OVER ADDITION:

For each  a,b,c in  \mathbb{R},  (a+b)\cdot c=ac+bc.

A trilogy of  (X,+,\cdot) satisfying the axioms I, II, III is called a field. According to this,  \mathbb{R} is a field.

IV. ORDER AXIOMS:

The relation " <" is defined over  \mathbb{R}. For each distinct  a,b in  \mathbb{R}, one and only one of the propositions  a<b and  b<a is true. By the help of the relation " <", the "at most" relation " \le" is defined as  a\le b  \Leftrightarrow  a<b or  a=b. Furthermore, the relation " <" satisfies the following properties:

IV {{}_{1}}.\,a<b and  b<c\Rightarrow a<c,

IV {{}_{2}}.\,a<b\Rightarrow \,\forall c\in \mathbb{R},\,a+c<b+c,

IV {{}_{3}}.\,a<b and  0<c\Rightarrow \,ac<bc.

 \mathbb{R} is a totally ordered set with the relation " \le".

V. COMPLETENESS AXIOM:

If it holds  \forall a\in A,  \forall b\in B,  a\le b for any two nonempty subsets  A and  B of  \mathbb{R}, then there exists a real number  c such as  \forall a\in A,  \forall b\in B ,  a\le c\le b.

All the properties of the real numbers except for I, II, III, IV, V can be easily proved by using the axioms I, II, III, IV, V. Now, we give some of them as a theorem:

THEOREM1:

1. In  \mathbb{R}, the additive identity element  0 is unique.

2. The additive inverse of each real number is unique. (The additive inverse of a real number  a is denoted by  -a. By the help of this, the substruction over  \mathbb{R} is defined by  a-b:= a+\left( -b \right))

3. For each  a,b in  \mathbb{R}, the unique solution of the equation  x+a=b is  x=b+(-a)=b-a.

4. In  \mathbb{R}, the multiplicative identity element  1 is unique.

5. The multiplicative inverse of each nonzero real number is unique. (The multiplicative inverse of a nonzero real number  a is denoted by  a^{-1} or  \displaystyle{\frac{1}{a}}. By the help of this, the fraction over  \mathbb{R} is defined by  \displaystyle{\frac{b}{a}}=b\cdot\frac{1}{a}:=b{{a}^{-1}}. Where  a is nonzero)

6. For each  a,b in  \mathbb{R} ( a\ne{0}), the unique solution of the equation  ax=b is  \displaystyle{x=b\cdot {{a}^{-1}}=b\cdot \frac{1}{a}=\frac{b}{a}}.

7.  \forall{a}\in \mathbb{R},  a\cdot 0=0.

8.  a\cdot b=0\,\Rightarrow \,a=0\lor{b=0}.

9.  \forall{a}\in \mathbb{R},  (-1)\cdot a=-a.

10.  \forall{a}\in \mathbb{R},  (-1)\cdot (-a)=a.

11.  \forall{a}\in \mathbb{R},  (-a)\cdot (-a)=a\cdot a={{a}^{2}}.

12. For any  a,b in  \mathbb{R},

 a<b\wedge b\le c\Rightarrow a<c,

 a\le b\wedge b<c\Rightarrow a<c.

13. For any  a,b,c,d in  \mathbb{R},

 0<a\Rightarrow -a<0,

 a\le b\wedge c\le d\Rightarrow a+c\le b+d,

 a\le b\wedge c<d\Rightarrow a+c<b+d.

14. For any  a,b,c in  \mathbb{R},

 0<a\wedge 0<b\Rightarrow 0<ab,

 a<0\wedge 0<b\Rightarrow ab<0,

 a<0\wedge b<0\Rightarrow 0<ab,

 a<b\wedge c<0\Rightarrow bc<ac.

15.  0<a\Rightarrow 0<{{a}^{-1}}.

16.  0<a\wedge a<b\Rightarrow 0<{{b}^{-1}}\wedge {{b}^{-1}}<{{a}^{-1}}.

PROOF:

The set of the real numbers satisfying the inequality  0<a (or  a>0) is called the positive real numbers and the set of the real numbers satisfying the inequality  a<0 is called the negative real numbers. They are denoted by  {{\mathbb{R}}^{+}}=\left\{ x\in \mathbb{R}\:|\:x>0 \right\} and  {{\mathbb{R}}^{-}}=\left\{ x\in \mathbb{R}\:|\:x<0 \right\}, respectively.

DEFINITION2: Let  X be a nonempty subset of  \mathbb{R}.

(i) If there exists a  b in  \mathbb{R} such that  \forall{x}\in{X}, x\le{b}, then  b is called an upper bound of  X.

(ii) If there exists an  a in  \mathbb{R} such that  \forall{x}\in{X}, a\le{x}, then  a is called an lower bound of  X.

(iii) If  X has an upper and a lower bound i.e.,  \exists{a,b}\in{\mathbb{R}}: a\le{x}\le{b}, then  X is called a bounded set.

(iv) If there exists an  M in  X such that  \forall{x}\in{X}, x\le{M}, then  M is called the maximum element of  X and denoted by

 M=\underset{x\in X}{\mathop{\max }}\,\left\{ x \right\} or  M=\max \left\{ x|x\in X \right\}

(v) If there exists an  m in  X such that  \forall{x}\in{X}, m\le{x}, then  m is called the minimum element of  X and denoted by

 m=\underset{x\in X}{\mathop{\min }}\,\left\{ x \right\} or  m=\min \left\{ x|\,x\in X \right\}

For example, the set  X=\left\{ x\in \mathbb{R}|\,-1<x<1 \right\} has no minimum and maximum elements. However, the set  Y=\left\{ x\in \mathbb{R}|\,-1\le x\le 1 \right\} has minimum and maximum elements. They are  -1 and  1, respectively.

When the set  X\subset \mathbb{R} has an upper bound, the set

 B=\left\{ b\in \mathbb{R}|\,\forall{x}\in{X}, x\le{b} \right\}

is nonempty. Similarly, when the set  X\subset \mathbb{R} has a lower bound, the set

 A=\left\{ a\in \mathbb{R}|\,\forall{x}\in{X}, a\le{x} \right\}

is nonempty.

DEFINITION3: Let  X be a nonempty subset of  \mathbb{R}.

(i) if  X has an upper bound, then the minimum of the set  B=\left\{ b\in \mathbb{R}|\,\forall{x}\in{X}, x\le{b} \right\} is called the supremum of  X and denoted by  \sup{X}.

(ii) if  X has a lower bound, then the maximum of the set  A=\left\{ a\in \mathbb{R}|\,\forall{x}\in{X}, a\le{x} \right\} is called the infimum of  X and denoted by  \inf{X}.

According to this definition, it holds,

 \sup{X}=\min \left\{ b\in \mathbb{R}|\,\forall{x}\in{X}, x\le{b} \right\},

 \inf{X}=\max \left\{ a\in \mathbb{R}|\,\forall{x}\in{X}, a\le{x} \right\}.

There may not be the minimum and the maximum of any nonempty subset of  \mathbb{R}. Is the property valid for the infimum and supremum? The following theorem is the answer of this question:

THEOREM2 (Least Upper Bound Property): Any subset of Real Numbers with an upper bound has unique supremum.

PROOF:

Similarly, the following theorem can be proved:

THEOREM3 (Greatest Lower Bound Property): Any subset of Real Numbers with an lower bound has unique infimum.

THEOREM4: Let  \varnothing\ne{X}\subset{\mathbb{R}} and  l,L\in{\mathbb{R}}. Then,

(i)  \sup{X}=L  \iff

(a)  \forall{x}\in{A}, x\le{L},

(b)  \forall{\varepsilon}>0,  \exists{x_{\varepsilon}}\in{X}:  L-\varepsilon<x_{\varepsilon}.

(ii)  \inf{X}=l  \iff

(a)  \forall{x}\in{X}, l\le{x},

(b)  \forall{\varepsilon}>0,  \exists{x_{\varepsilon}}\in{X}:  x_{\varepsilon}<l+\varepsilon.

PROOF:

DEFINITION4: Let  a and  b be two real numbers and  a<b. The set  \left\{ x\in \mathbb{R}\,|\,a<x<b \right\} is called an open interval with the endpoint  a and  b and denoted by  \left( a,b \right) (or  \left] a,b \right[). Similarly, the set  \left\{ x\in \mathbb{R}\,|\,a\le x\le b \right\} is called a closed interval with the endpoint  a and  b and denoted by  \left[ a,b \right]. A left-open right-closed interval and a left-closed right-open interval are defined by the follows:

 \left( a,b \right]=\left] a,b \right]=\left\{ x\in \mathbb{R}\,|\,a<x\le b \right\} left-open right-closed interval

 \left[ a,b \right)=\left[ a,b \right[=\left\{ x\in \mathbb{R}\,|\,a\le x<b \right\} left-closed right open interval

For  a and  b in  \mathbb{R}, the unbounded intervals  (a,+\infty),  [a,+\infty),  (-\infty,b),  (-\infty,b] and  (-\infty,+\infty) are defined by the follows:

 (a,+\infty)=\left\{ x\in \mathbb{R}\,|\,x>a \right\},

 [a,+\infty)=\left\{ x\in \mathbb{R}\,|\,x\ge a \right\},

 (-\infty,b)=\left\{ x\in \mathbb{R}\,|\,x<b \right\},

 (-\infty,b]=\left\{ x\in \mathbb{R}\,|\,x\le b \right\},

 (-\infty,+\infty)=\mathbb{R}.

The set obtained by adding two elements  -\infty and  +\infty read as "negatif infinity" and "positive infinity" to  \mathbb{R} is called the extended real number line or the extended real number system and denoted by  \overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}. Henceforth, we assume that the extended real number system satisfies the following properties:

1.  \forall x\in \mathbb{R},

a)  -\infty <x<+\infty,

b)  x-(+\infty )=x-\infty =-\infty,

c)  x+(+\infty )=x+\infty =+\infty,

d)  x-(-\infty )=x+\infty =+\infty,

2.

a)  +\infty +(+\infty )=+\infty,

b)  -\infty +(-\infty )=-\infty,

3.  \forall x\in {{\mathbb{R}}_{+}},

a)  x(+\infty )=+\infty,

b)  x(-\infty )=-\infty,

4.  \forall x\in {{\mathbb{R}}_{-}},

a)  x(+\infty )=-\infty,

b)  x(-\infty )=+\infty,

5.

a)  (+\infty )(+\infty )=+\infty,

b)  (-\infty )(-\infty )=+\infty,

c)  (+\infty )(-\infty )=-\infty,

6.  \forall x\in \mathbb{R},

a)  \displaystyle{\frac{x}{+\infty }=0},

b)  \displaystyle{\frac{x}{-\infty }=0}.

Let  X be a nonempty subset of  \overline{\mathbb{R}}. If  X has no a lower bound, then the infimum is defined as \inf X=-\infty and if  X has no an upper bound, then the supremum is defined as  \sup X=+\infty. According to this, each nonempty subset of  \overline{\mathbb{R}} has both the infimum and the supremum.

DEFINITION5: The absolute value or modulus of a real number  x is defined as follows:

 |x|=\left\{ \begin{array}{rl} x, & \text{if } x\ge{0}\\ -x, & \text{if }x<0 \end{array} \right.

THEOREM5:

a)  \forall{x}\in{\mathbb{R}},  |-x|=|x|\ge{0},

b)  |x|=0\Leftrightarrow{x=0},

c)  \forall{x}\in{\mathbb{R}},  -x\le{|x|} and  x\le{|x|},

d)  \forall{a,b}\in{\mathbb{R}},  |ab|=|a||b| and  \displaystyle{\left| \frac{a}{b} \right| =\frac{|a|}{|b|}}  (b\ne{0}),

e)  \forall{a,b}\in{\mathbb{R}},  |a\pm{b}|\le{|a|+|b|},

f)  \forall{a,b}\in{\mathbb{R}},  \big| |a|-|b| \big| \le{|a-b|},

g)  |x|<r\Leftrightarrow{-r<x<r},

h)  |x|\le{r}\Leftrightarrow{-r\le{x}\le{r}},

i)  |x|>r\Leftrightarrow{x<-r\lor{x>r}},

j)  |x|\ge{r}\Leftrightarrow{x\le{-r}\lor{x\ge{r}}}.

PROOF:

For two real numbers  a,b, the number  \left| a-b \right|=\left| b-a \right| is called the distance between  a and  b and denoted by  d(a,b).

Let  a and  b be two real numbers and  a<b. The number  b-a>0 is called the length, width, measure or size of the intervals  (a,b),  [a,b),  (a,b] and  [a,b].

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