Construction of The Real Numbers

On December 12, 2010, in Analysis, by ufukkaya

DEFINITION1: The set $\mathbb{R}$ having 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 ${{}_{1}}.\,a<b$ ve $b<c\Rightarrow a<c$ ,

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

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

$\mathbb{R}$ 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: (will be added)

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): There exists the supremum of any nonempty subset of $\mathbb{R}$ having an upper bound.

PROOF: (will be added)

Similarly, the following theorem can be proved:

THEOREM3 (Greatest Lower Bound Property): There exists the infimum of any nonempty subset of $\mathbb{R}$ having a lower bound.

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: (will be added)

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

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

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:

$|x|=\bigg\{$ $x,\:\:\:\:\:\text{if}\:x\ge{0}\\-x,\,\text{if}\: x<0$

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 (will be added)

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 $[a,b]$ .

Academic Maths

Construction of The Real Numbers

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