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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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zhym@fudan.edu.cn

BBS id:abchjsabc 软件楼103

BBS id:chengyiting

liy@fudan.edu.cn李弋

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,
Example：R={(a,b),(b,a),(a,c)},is not symmetric
• + (c,a),R\'={(a,b),(b,a),(a,c), (c,a)}，R\'is symmetric.
• Closure
2.5 Closures of Relations
• 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
• (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\'\'
• 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 2.9: Let R be a relation on A. Then

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

2.6 Equivalence Relation

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?)

2.Equivalence classes

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:P146 32,34
• P151 1,2,13, 17, 23,24
• P167 15,16,22,24,26,27,28,29,32, 36