Definition of relation equations Quv◦Xvw=Suw in the set of finite natural numbers is given. Rapid method of solving the maximal solution of relation equations Quv◦Xvw=Suw in the set of finite natural numbers is provided

DONG, Xiao

LUO, Yaling

ZHANG, Shiqiang

English

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ISSN 1925-251X [Print]ISSN 1925-2528 [Online]www.cscanada.netwww.cscanada.orgProgress in Applied Mathematics Vol. 8, No. 1, 2014, pp. 17-20DOI: 10.3968/579517Relation Equations in the Set of Finite Natural Numbers and Its Maximal SolutionZHANG Shiqiang[a] ; DONG Xiao[a] ; LUO Yaling[a] ,* [a]Medical Information College, Chongqing Medical University,  Chongqing, China.* Corresponding author. Address: Medical Information College, Chongqing Medical University, 400016, China. E-mail: math808543@sina.comReceived March 14, 2014/ accepted June 5, 2014/ Published online July 26, 2014Abstract：Definition of relation equations Quv◦Xvw=Suw in the set of finite nat-ural numbers is given. Rapid method of solving the maximal solution of relation equations Quv◦Xvw=Suw in the set of finite natural numbers is provided.Key words: Set of finite natural numbers; Relation equations; Maximal solutionZhang, S. Q., Dong, X., & Luo, Y. L. (2013). Relation Equations in the Set of Finite Natural Numbers and Its Maximal Solution. Progress in Applied Mathematics, 8(1), 17-20. Available from http://www.cscanada.net/index.php/pam/article/view/5795 DOI: http://dx.doi.org/10.3968/57951. INTRODUCTIONN={0, 1, 2,…} is a set of natural numbers[1]. Nn*={0, 1, 2,…, n*} is a set of finite natural numbers, among them, n* is an arbitrary natural number larger than zero. Starting from Jieshengjishi, human ancestors began to use a set of finite natural numbers. Since the discovery of the set of natural numbers, the set of rational numbers, the set of real numbers, the set of finite natural numbers seem to fade out of sight. The emergence of computer, and the line of sight of people back to the set of finite natural numbers. In the computer, the set {0, 1} is the most simple set of finite natural numbers[2]. In the image processing, the set {0, 1, 2,…, 255} is the set of finite natural numbers[3].The relation equation on the set of natural numbers has been discussed very mature, but the relation equation on the set of finite natural numbers is not mentioned. One of the reasons is that conventional operators on the set of natural numbers does not satisfy closed on the set of finite natural numbers. But that information processing dependent the set of finite natural numbers show importance of relation equations on the set of finite natural numbers.Relation Equations in the Set of Finite Natural Numbers and Its Maximal Solution182.  OPERATORS AND RELATION EQUATIONON IN THE SET OF FINITE NATURAL NUMBERSAddition and multiplication on elements in the set Nn*={0, 1, 2,…, n*} of finite natural numbers is defined as follows:Definition 1 For given element m∈Nn* and element n∈Nn*, multiplication is defined as follows:    m×n=min{m, n}= m∧n . (1)Definition 2 For given element m∈Nn* and element n∈Nn*, addition is defined as follows:    m+n=max{m, n}= m∨n . (2)Definition 3 With F(Nn*) representative power set in the set Nn*={0, 1, 2,…, n*} of finite natural numbers, for given set U={u1, u2,…, um}∈F(Nn*) and V={ v1, v2,…, vn}∈F(Nn*), the product set of U and V is defined as followsP=U×V={p(u,v)|u∈U,v∈V}among them, 0≤u1≤ u2≤…≤um≤n*  and  0≤v1≤ v2≤…≤vn≤n*.Definition 4 With F(U×V) representative all product set on the U and V, for given product set Q={q(u,v)|u∈U, v∈V}∈F(U×V), X={x(u,v)|u∈U, v∈V}∈F(V×W) and S={s(u,v)|u∈U, v∈V}∈F(U×W), relationship equations for opertor (∧,∨)in the set Nn*={0, 1, 2,…, n*} of finite natural numbers is defined as follows:    Quv◦Xvw=Suw , (3)among them  )]([][ , vwwuuvvuwuw rqsS ∧∨== ≤>=nmnnmnnm *α <≥=nmnmnnm0β )]([][, vwwuuvvuwuwrqsS α∧== )]([][,vwwuuvvuwuwrqsS β∨== )]([, uwwvvuusq α∧.Definition 5 For given element m∈Nn* and element n∈Nn*, operator α is defined as follows:     )]([][, vwwuuvvuwuwrqsS ∧∨== ≤>=nmnnmnnm *α <≥=nmnmnnm0β )]([][, vwwuuvvuwuwrqsS α∧== )]([][,vwwuuvvuwuwrqsS β∨== )]([, uwwvvuusq α∧. (4)Definition 6 For given element m∈Nn* and element n∈Nn*, operator β is defined as follows:     )]([][, vwwuuvvuwuwrqsS ∧∨== ≤>=nmnnmnnm *α <≥=nmnmnnm0β )]([][, vwwuuvvuwuwrqsS α∧== )]([][,vwwuuvvuwuwrqsS β∨== )]([, uwwvvuusq α∧. (5)Definition 7 For given product set Q={q(u,v)|u∈U,v∈V}∈F(U×V),R={r(u,v)|u∈U,v∈V}∈F(V×W), opertoration for opertor (α,∧) in the set Nn*={0, 1, 2,…, n*} of finite natural numbers is defined as follows:    S=Q＠R∈F(U×W) , (6)among them  )]([][, vwwuuvvuwuwrqsS ∧∨== ≤>=nmnnmnnm *α <≥=nmnmnnm0β )]([][, vwwuuvvuwuwrqsS α∧== )]([][,vwwuuvvuwuwrqsS β∨== )]([, uwwvvuusq α∧.Def in i t i on  8  For  g iven  p roduc t  s e t  Q={q (u ,v ) |u∈U ,v∈V}∈F (U×V ) , R={r(u,v)|u∈U,v∈V}∈F(V×W), opertoration for opertor (β,∨) in the set Nn*={0, 1, 2,…, n*} of finite natural numbers is defined as follows:Zhang, S. Q., & Dong, X., & Luo, Y. L./Progress in Applied Mathematics, 8(1), 201319    S=Q＆R∈F(U×W) , (7)among them  )]([][, vwwuuvvuwuwrqsS ∧∨== ≤>=nmnnmnnm *α <≥=nmnmnnm0β )]([][, vwwuuvvuwuwrqsS α∧== )]([][,vwwuuvvuwuwrqsS β∨== )]([, uwwvvuusq α∧ .3.  THE MAXIMUM SOLUTION OF THE RELATION EQUATIONON IN THE SET OF FINITE NATURAL NUMBERSTheorem 1. If the relation equation Quv◦Xvw=Suw in the set Nn*={0, 1, 2,…, n*} of finite natural numbers has solutions, then    [QT＠Suw]= )]([][, vwwuuvvuwuwrqsS ∧∨== ≤>=nmnnmnnm *α <≥=nmnmnnm0β )]([][, vwwuuvvuwuwrqsS α∧== )]([][,vwwuuvvuwuwrqsS β∨== )]([, uwwvvuusq α∧  (8)is the maximum solution of the relation equation Quv◦Xvw=Suw in the set Nn*={0, 1, 2,…, n*} of finite natural numbers.Proof. If the relation equation Quv◦Xvw=Suw in the set Nn*={0, 1, 2,…, n*} of finite natural numbers has solutions, let qTuv=qvu, thenSuw=Quv◦Rvw⊆ Quv◦[QT＠Suw] = ])][([,, twwvTtvUtwuuvvsqq α∈∧∧∨  ]][[ **,,(*)wuwvTvuwuuvvsqq α∧∨≤  = ]][[ **,, wuwvvuwuuvvsqq α∧∨  = ][ *, wuwuuvvsq ∧∨  uwuwuwvSss ==∨≤ ][][ .Quv◦[QT＠[Quv◦Rvw]]= Quv◦[QT＠Suw] .Because⊆ Quv◦[QT＠Suw] = ])][([,, twwvTtvUtwuuvvsqq α∈∧∧∨  ]][[ **,,(*)wuwvTvuwuuvvsqq α∧∨≤  = ]][[ **,, wuwvvuwuuvvsqq α∧∨  = ][ *, wuwuuvvsq ∧∨  uwuwuwvSss ==∨≤ ][][ .That is     Suw=Quv◦Rvw⊆ Quv◦[QT＠Suw] = ])][([,, twwvTtvUtwuuvvsqq α∈∧∧∨  ]][[ **,,(*)wuwvTvuwuuvvsqq α∧∨≤  = ]][[ **,, wuwvvuwuuvvsqq α∧∨  = ][ *, wuwuuvvsq ∧∨  uwuwuwvSss ==∨≤ ][][ .Quv◦[QT＠Suw] ⊆ Quv◦[QT＠Suw] = ])][([,, twwvTtvUtwuuvvsqq α∈∧∧∨  ]][[ **,,(*)wuwvTvuwuuvvsqq α∧∨≤  = ]][[ **,, wuwvvuwuuvvsqq α∧∨  = ][ *, wuwuuvvsq ∧∨  uwuwuwvSss ==∨≤ ][][ . Suw  . (9)The above Formula (9) shows that [QT＠Suw] is the solution of the relation equation Quv◦Xvw=Suw in the set Nn*={0, 1, 2,…, n*} of finite natural numbers.Let Rvw is another solution of the relation equation uv◦Xvw=  in the set Nn*={0, 1, 2,…, n*} of finite natural numbers, then[QT＠Suw]= [QT＠(Q◦Rvw)] ])[([,, vwwuuvvwvTuvurqq ∧∨∧= α  ][[,,(*)vwwuuvwvvuurqq ∧∧≥ α  ][][(*)vwvwurr =∧≥  . ⊇That is     [QT＠Suw][QT＠Suw]= [QT＠(Q◦Rvw)] ])[([,, vwwuuvvwvTuvurqq ∧∨∧= α  ][[,,(*)vwwuuvwvvuurqq ∧∧≥ α  ][][(*)vwvwurr =∧≥  . ⊇ Rvw . (10)The above Formula (10) shows that another solution Rvw of the relation equation Quv◦Xvw=Suw in the set Nn*={0, 1, 2,…, n*} of finite natural numbers less than  the solution Relation Equations in the Set of Finite Natural Numbers and Its Maximal Solution20[QT＠Suw] of the relation equation Quv◦Xvw=Suw in the set Nn*={0, 1, 2,…, n*} of finite natural numbers.So that the solution [QT＠Suw] is the Maximum solution of the relation equation Quv◦Xvw=Suw in the set Nn*={0, 1, 2,…, n*} of finite natural numbers.4. CONCLUSIONBecause that information processing dependent the set of finite natural numbers show importance of relation equations in the set of finite natural numbers. This paper gives definition of relation equations in the set of finite natural numbers. At the same time the papre provids a method of solving the maximal solution of relation equations in the set of finite natural numbers. In the future, we will discuss the solving method of solutions of relation equations in the set of finite natural numbers and will discuss other properties of the relation equations in the set of finite natural numbers.REFERENCES[1] Hua, L. G. (1963). Advanced mathematics. Beijing, China: Science Press.[2] Gibas, C., & Jambeck, P. (2001). Developing bioinformatics computer skills. Cambridge, England: O Reilly＆Associates, 1nc.[3] Liu, J. (2003). Digital image processing and advanced applications. Beijing, China: Science Press.[4] Hua, L. G. (1957). Number theoretic guidance. Beijing, China: Science Press.[5 Cheng, Y. Y. (1984). Fuzzy mathematics (pp.30-36). Wuhan, China: Huazhong Institute of Technology Press.[6] Sanchez, E. (1976). Resolution of composite fuzzy relation equations. Information and Control, 30, 38-48.

Relation Equations in the Set of Finite Natural Numbers and Its Maximal Solution

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Relation Equations in the Set of Finite Natural Numbers and Its Maximal Solution

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