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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