Applications Of Graph Theory In Set Theory

Graph theory is a component of mathematics, which is mainly about the study of graphs. The graph in graph theory consists of a set of points together with lines joining certain pairs of these points. This kind of graph is usually used to describe certain relations of some things, in which points stand for things and lines stand for the relations between the things.Graph theory is a part of applied mathematics indeed, thus in history it has been established by many mathematicians respectively. The earliest known paper on graph theory appeared in the works of Euler in 1736, which considered the primitive problems that had strong practical background.Graph theory derived from famous problem of the seven bridges of Konigsberg (see [1]). Euler solved this problem in 1736.He changed this problem into the first graph theory problem by abstract analyzed method. Euler proved that this problem had no solutions. Moreover, he extended this problem. This job made Euler become the sponsor of graph theory (and topology).The extensive applications of graph theory have promoted its own development. From 1940s to 1960s some theories about graphs such as matroid theory, hyper graph theory, polar graph theory, algebraic graph theory and topological graph theory (see [2-8])had developed greatly.Set theory has become a subject since the late 19th century. From 1874 to 1897 the sponsor of the theory G. Cantor proposed a series of these that gave the foundation of set theory. Since then, the concepts and results of the theory have been applied in almost every component of mathematics, which have influenced mathematics deeply.The outcome of Cantor's set theory gave rise to great reflection in the field ofmathematics at that time. It was approved and praised greatly by some mathematicians such as R. Dedekind, B. Russell, D. Hilbert, and so on. It was also strongly opposed by some mathematicians especially L.Kronecker. Since the late of 19th century, various paradoxes had appeared continually and at that time such paradoxes couldn't be answered applying set theory.In order to remedy Cantor's theory, in 1908 E. Zermelo gave the first axiom of set theory(see [9] [10]). These axioms make it clear that what is reasonable to deal with sets. After complemented and consummated by other mathematicians including A. Fraenkel, the ZF axiomatic system appeared followed by GB axiomatic system later that was given by others such as von Newmann, P. Bernays, K. Godel (see [11,12,13,14]). Such systems excluded the paradoxes and weakened the voice of critics. From then on, the research of the choice axioms and continuum hypothesis have made set theory develop greatly and become one of the dynamic mathematical subjects.We take advantages of good positions of graph theory to give some conclusions and their proofs about relations in set theory in this paper.This paper is divided into four chapters as follows:Chapter 1: Preface.Chapter 2: Some definitions about well structured graphs and relations and the following theorem with its proof are given in this chapter.Theorem 2.1(AC) Graph D(V,A) is well structured ifand only if there isn't a decreasing a- sequence, that is, there isn't a sequence (x/Beo) such thatChapter 3: Some definitions about finitely compound graphs and the following theorems are given in this chapter.Theorem 3.5 The finitely compound relation R* of the relation R is transmittable and it is the smallest transmittable expansion of R.Collary3.2 R'=R ifand only if R is transmittable.Theorem 3.6 The finitely compound relation R' of the relation R is wellfounded if and only if the relation R is well founded.Theorem 3.7 The finitely compound relation R~l* of the reversal relation R~l of the relation/? is the same as the reversal relation R'~lof the finitely compound relation R', that is, J?"'*=i?M.Chapter 4.: Some definitions about trichotomy of relations of graphs and the following theorems are given using the concepts appeared in the former chapters.Theorem 4.3 If a well-structured graph D(V,A) is A joined, then it has A trichotomy.Theorem 4.5 The finitely compound graphD(V, A'} of a single-side connected graphD(V,A) is A* joined.Theorem 4.6 A graph D(V, A) is single-direct connected graph if and only if its finitely compound graph D (V, A*) has A* trichotomy.Theorem 4.7 If a relation has trichotomy, and then it is not reversed.Theorem 4.8 If a relation if is single-direct connected, and then its finitely compound R' is not reversed.Theorem 4.9 For a well-founded relation .R, it is joined if and only it has trichotomy.Collary4.3 If a relation R is joined, then its finitely compound R* has trichotomy.Theorem 4.10 A relation R is single-direct connected if and only if its finitely compound R* has trichotomy.The main creative points of this paper are as follows:1: Some conclusions about well-founded relations are given taking advantage of the concepts about well-structured graphs.2: Some theories about certain special relations are established taking good use of finitely compound graphs.3: Using some concepts about trichotomy and connection of graphs approaches some properties about relations.