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# 第 3 周起每周一交作业，作业成绩占总成绩的1 5% ； 平时不定期的进行小测验，占总成绩的 15% ； 期中考试成绩占总成绩的 20% ；期终考试成绩占总成绩的 50% - PowerPoint PPT Presentation

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3周起每周一交作业，作业成绩占总成绩的15%；

BBS id:abchjsabc 软件楼103

BBS id:chengyiting

• 2.Composition

• Definition 2.14: Let R1 be a relation from A to B, and R2 be a relation from B to C. The composition of R1 and R2, we write R2R1, is a relation from A to C, and is defined R2R1={(a,c)|there exist some bB so that (a,b)R1 and (b,c)R2, where aA and cC}.

• (1)R1 is a relation from A to B, and R2 is a relation from B to C

• (2)commutative law? 

• R1={(a1,b1), (a2,b3), (a1,b2)}

• R2={(b4,a1), (b4,c1), (b2,a2), (b3,c2)}

• Associative law?

• For R1A×B, R2B×C, and R3C×D

• R3(R2R1)=?(R3R2)R1

• subset of A×D

• For any (a,d)R3(R2R1), (a,d)?(R3R2)R1,

• Similarity, (R3R2)R1R3(R2R1)

• Theorem 2.3：Let R1 be a relation from A to B, R2 be a relation from B to C, R3 be a relation from C to D. Then R3(R2R1)=(R3R2)R1(Associative law)

• Definition 2.15: LetR be a relation on A, and nN. The relation Rn is defined as follows.

• (1)R0 ={(a,a)|aA}), we write IA.

• (2)Rn+1=RRn.

• Theorem 2.4: LetR be a relation on A, and m,nN. Then

• (1)RmRn=Rm+n

• (2)(Rm)n=Rmn

• A={a1,a2,,an},B={b1,b2,,bm}

• R1 and R2 be relations from A to B.

• MR1=(xij), MR2=(yij)

• MR1∪R2=(xijyij)

• MR1∩R2=(xijyij)

•  0 1  0 1

• 0 0 1 0 0 0

• 1 1 1 10 1

• Example:A={2,3,4},B={1,3,5,7}

• R1={(2,3),(2,5),(2,7),(3,5),(3,7),(4,5),(4,7)}

• R2={(2,5),(3,3),(4,1),(4,7)}

• Inverse relation R-1 of R : MR-1=MRT, MRT is the transpose of MR.

• A={a1,a2,,an},B={b1,b2,,bm}, C={c1,c2,,cr},

• R1 be a relations from A to B, MR1=(xij)mn, R2 be a relation from B to C, MR2=(yij)nr. The composition R2R1 of R1 and R2,

• 1.Introduction

• Constructa new relation R‘,s.t. RR’,

• particular property,

• smallest relation

• closure

• Definition 2.17: Let R be a relation on a set A. R' is called the reflexive(symmetric, transitive) closure of R, we write r(R)(s(R),t(R) or R+), if there is a relation R' with reflexivity (symmetry, transitivity) containing R such that R' is a subset of every relation with reflexivity (symmetry, transitivity) containing R.

• Condition:

• 1)R' is reflexivity(symmetry, transitivity)

• 2)RR'

• 3)For any reflexive(symmetric, transitive) relation R", If RR", then R'R"

• Example：If R is symmetric, s(R)=?

• If R is symmetric，then s(R)=R

• Contrariwise, If s(R)=R，then R is symmetric

• R is symmetric if only if s(R)=R

• Theorem 2.5: Let R be a relation on a set A. Then

• (1)R is reflexive if only if r(R)=R

• (2)R is symmetric if only if s(R)=R

• (3)R is transitive if only if t(R)=R

• Theorem 2.6: Let R1 and R2 be relations on A, and R1R2. Then

• (1)r(R1)r(R2)；

• (2)s(R1)s(R2)；

• (3)t(R1)t(R2)。

• Proof: (3)R1R2t(R1)t(R2)

• Because R1R2， R1t(R2)

• t(R2) :transitivity

• Example:Let A={1,2,3},R={(1,2),(1,3)}. Then

• 2.Computing closures

• Theorem 2.7: Let R be a relation on a set A, and IA be identity(diagonal) relation. Then r(R)=R∪IA(IA={(a,a)|aA})

• Proof：Let R'=R∪IA.

• Definition of closure

• (1)For any aA, (a,a)?R'.

• (2) R?R'.

• (3)Suppose that R'' is reflexive and RR''，R'?R''

• Theorem 2.8：Let R be a relation on a set A. Then s(R)=R∪R-1.

• Proof：Let R'=R∪R-1

• Definition of closure

• (1) R', symmetric?

• (2) R?R'.

• (3)Suppose that R'' is symmetric and RR''，R'?R'')

• Example ：symmetric closure of “<” on the set of integers,is“≠”

• <,>，

• Let A is no empty set.

• The reflexive closure of empty relation on A is the identity relation on A

• The symmetric closure of empty relation on A, is an empty relation.

Theorem 2.10：Let A be a set with |A|=n, and let R be a relation on A. Then

• Example：A={a,b,c,d},R={(a,b),(b,a), (b,c),(c,d)},t(R)=?

1.Equivalence relation

Definition 2.18: A relation R on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.

Example: Let m be a positive integer with m>1. Show that congruence modulo m is an equivalence relation. R={(a,b)|ab mod m}

Proof: (1)reflexive (for any aZ，aRa?)

(2)symmetric (for any aRb， bRa?)

(3)transitive (for aRb，bRc，aRc?)

partition

Definition 2.19: A partition or quotient set of a nonempty set A is a collection  of nonempty subsets of A such that

(1)Each element of A belongs to one of the sets in .

(2)If Ai and Aj are distinct elements of , then Ai∩Aj=.

The sets in  are called the bocks or cells of the partition.

Example: Let A={a,b,c},

P={{a,b},{c}},S={{a},{b},{c}},T={{a,b,c}},

U={{a},{c}},V={{a,b},{b,c}},W={{a,b},{a,c},{c}},

infinite

Example：congruence modulo 2 is an equivalence relation.

For any xZ, or x=0 mod 2,or x=1 mod 2, i.e or xE ,or xO.

And E∩O=

E and O，

{E, O} is a partition of Z

Definition 2.20: Let R be an equivalence relation on a set A. The set of all element that are related to an element a of A is called the equivalence class of a. The equivalence class of a with respect to R is denoted by [a]R, When only one relation is under consideration, we will delete the subscript R and write [a] for this equivalence class.

Example：equivalence classes of congruence modulo 2 are [0] and [1]。

[0]={…,-4,-2,0,2,4,…}=[2]=[4]=[-2]=[-4]=…

[1]={…,-3,-1,1,3,…}=[3]=[-1]=[-3]=…

the partition of Z =Z/R={[0],[1]}

• Exercise: A. The set of all element that are related to an element P146 32,34

• P151 1,2,13, 17, 23,24

• P167 15,16,22,24,26,27,28,29,32, 36