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:


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

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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