2014.7.3 二元关系.ppt
《2014.7.3 二元关系.ppt》由会员分享,可在线阅读,更多相关《2014.7.3 二元关系.ppt(141页珍藏版)》请在得力文库 - 分享文档赚钱的网站上搜索。
1、1,The fourth chapter binary relation,Relationship:Relationships of people have friends relationship,parent-child relationships, parent-child relationships, classmates. There are greater than, less than, equal to relations, and divisible, congruence relations between two figures. Two variables, such
2、as function relationship.Relationship of mathematical concept is to establish the concept of relationships in daily life.,2,eg suppose set A=a,b,c,d is made up from male team membersone of one table-tennis team. Set B=e,f,g is made up from female team membersone . If there are mixed doubles match re
3、lationship between A and B ,stant for a and g,d and e。so can be expressed as: R=,“R” is stant for the sequence pair set which is have mixed doubles match relationship.All the sequence pair set that may have the mixed doubles match relationship are: AB=|xAyB =,,3,There are people A,B,C,and four opera
4、ting post,. A can take up workand ,Bcan take up work ,C can take up work and。So, the relatioship of people and work can be expressed as: R=, This is the relationship between setA,B,C and set,.,4,4.1 Relationship and its representation,一、Definition of relationship(1) The subset of AB can be seen as a
5、 binary relation A to B.(2) The subset of A1A2An(n1) can be seen as a n-ary relation about A1A2An.,(3)The subset of can be seen as a n-ary relation about A.From the concept can know relationship is a set.The methodused to definit set is appropriate for relationship.,5,For example,the binary relation
6、 about real number R can be definit as follow: =|xRyRxyThe most important is binary relation.This chapter mainly discuss the binary relation,The term relationship refers to the binary relation.Binary relations have their own special notation and several new terms.Otherwise will be noted especially.,
7、6,suppose set A=x1,x2,x7, B=y1,y2,y6 R=,obvious R,can be wright as x3Ry1,also called infix notation:x3 and y1 have the relation R.infix notation used to expressed the relationship of “=”,“”,“”and so on,for example ,can be wright as 35.A can be seen as the former domain of relationship R,B can be see
8、n as codomain of the relationship R. D(R)=x|y(R) can be seen as domain of definition of relationship R. R(R)=y|x(R)can be seen as range of relationship R.,7,Relationship is a set of ordered pair,and can be operat as well as set.The result definit a new relationship.Suppose R and S are two binary rel
9、ation about the given set.Then RS、RS、R-S,R can be definited as follow: x(RS)y xRyxSy x(RS)y xRyxSy x(R-S)y xRyx$y x(R)y x R y,8,Thought:if |A|=m,|B|=n ,how many relationship exist from A to B?,solve:Due to the relation is a subset of AB.So the subset of AB are 2mn, according to the conclusion of the
10、 number of power set.So ,there are 2mn relations from A to B.specially ,when A=B,then there are 2n2 relations about A.,9,二、Some special relations For arbitrary set A there are three kinds of special relationships: empty relation、Global relationshipUA和 identity relation A.Definition:For arbitrary set
11、 A,there are three special subset about AA:(1) is a subset of AA,from the definition of on A,called the empty relation on A.(2) AA itself is a subset AA,from the definition of AAabout A,called Global relationship about A. denote as UA, UA=aAbA(3)A=aA was called identity relation about A.,10,eg suppo
12、se setA=0,1,2,then:UA=,Due to UAis global relations upon A, obvious UA=AA=9 A=,If is A identity relation upon A,then A=A=3,11,三、Example for commonly relationeg1 Less than or equal to relationship.A=1,2,3,definition the relation upon A LA=a,bAab。then LA =,eg2 Aliquot relationship.suppose set B=1,2,3,
13、4,5,6,definition the binary relation upon B DB=a,bBbaN.then:DB=,,,attention:baN,in other word is a can exact division b,or b can be exacted division by a.,12,eg3 congruence relation A=1,2,3,4,the binary relation upon set A is: R=( a-b)/2Za,bA。then R=, Description:The relation is called “module2 cong
14、ruence relation”,denoted a=b(mod2).,13,eg 4 inclusion relation suppose set A,define R is a relation upon (A), R=u(A),v(A),and uv。eg,suppose set A=a,b, (A)= ,a,b,A,thenR=,,14,eg 5 suppose set A=2,3,4,5,6,Express the following relationship with enumeration method:R=a is the integer times of bR=(a-b)2A
15、R=a/b is prime number R=ab R=a,b coprime numbers,15,solve : because set A=2,3,4,5,6,R=a is the integer times of b。R=, R=(a-b)2A。R=, R=a/b is prime number。R=, R=ab。R=EA-A,there are 25-5 ordered pairs. R=a,b coprime numbersR=,,,,16,四、Representation of the relation The relation that have been arisen pr
16、evious are defined with the definition of set. We can express the binary relation of finite set with matrix of relation and relation diagrams.Definition: suppose set A=a1, ,am,B=b1,bnR is a relation of A to B,then the matrix of relation of R is a mn-matrix MR=(rij)mn rij =1,when R rij =0,when Rif R
17、is a relation upon set A, then the matrix of relation is a square matrix,17,eg set A=a,b,c,d, B=x,y,z, R=, Due to |A|=4,|B|=3, then MR is 43 matrix: eg A=2,3,4,5,6,then the matrix of relation of R1=|a is the integer times of b is :,18,Related description: All of the elements of the matrix of relatio
18、n M of empty relation are 0. All of the elements of the matrix of relation MU of global relationship UA are 1. All the diagonal elements of matrix of relation MI of identity relation A are 1, undiagonal elements are 0 .The matrix is called unit matrix in linear algebra, and denoted .,19,Definition s
19、uppose set A=a1,am,B=b1,bn,(AB) R is a relation of A to B,the amounts of circle is m+n, respectively represent a1,am and b1,bn(On both sides respectively),the circles are called knot.If R,then connect a directed edge (arc)from ai to bj , if R,then do not connect the corresponding edge (arc).This gra
20、ph is called relational graph of relation R. If R is a relation upon A, then only make m circles represent a1,am,(Dot not make 2m circles,and No longer on both sides respectively ) directed edge (arc),the mathod is same as the ,if R,then make a directed arc from ai to ai,this arc is called self-cont
21、ained loop.,20,eg 1 A=a,b,c,d, B=x,y,z, R=, The following is relational graph of relation R.,21,eg2 suppose setA=1,2,3,4,R=, is a relation upon A.Then the matrix of relation and relational graph of R :In the relational graph, the vertices is 1,2,3,4 ,From ,R can know,there is an edge from 1 to 1 in
22、the graph ,(the loopthrough 1a ring),There is one edge from 1 to 2. The others edges of ordered pairs can be made similarly.,22,eg3 suppsose set A=2,3,4,5,6,draw up the matrix of relation and relational graph of the following relation: R=A multiple of a is b.R=(a-b)2A.R=a/b is prime number.R=a,brela
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 2014.7 二元关系
限制150内