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

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Partial Order Relation

On September 18, 2010, in Analysis, by ufukkaya

DEFINITION1: Let $X$ be a set and $R\subset{X\times{X}}$ . If the relation $R$ is reflexive, antisymmetric and transitive, then the relation $R$ is called a “partial order relation” and denoted by $R=\le$ in general. If “ $\le$ ” is a partial order relation over a set $X$ , then $(X,\le)$ is called “partially ordered set” or shortly “poset”.

DEFINITION2: Let $x$ and $y$ are elements of a partially ordered set $X$ . If it holds “ $x\le{y}\lor{y\le{x}}$ ”, then $x$ and $y$ are called “comparable”. Otherwise they are called “incomparable”.

DEFINITION3: If $x$ and $y$ are comparable for all $x,y$ in a partially ordered set $(X,\le)$ , then the relation $\le$ is called a “total order” and the set $X$ is called a “totally ordered set” or “linearly ordered set”.

DEFINITION4: Let $(X,\le)$ be a partially ordered set and $A\subset{X}$ . If $(A,\le)$ is a totally ordered set, then $A$ is called a “chain” in $X$ .

DEFINITION5: Let $(X,\le)$ be a partially ordered set and $A\subset{X}$ . If there exists an element $a^{*}\in{A}$ satisfying $a\le{a^{*}}$ for all $a\in{A}$ , then $a^{*}$ is called the maximum of $A$ , and if there exists an element $a_{*}\in{A}$ satisfying $a_{*}\le{a}$ for all $a\in{A}$ , then $a_{*}$ is called the minimum of $A$ . The minimum and the maximum of $A$ are denoted by $\min{A}$ and $\max{A}$ respectively.

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