每周三交作业，作业成绩占总成绩的1 5% ；...

每周三交作业，作业成绩占总成绩的 15% ；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%zhym@fudan.edu.cn张宓 13212010027@fudan.edu.cnBBS id:abchjsabc 软件楼 1039杨侃 10302010007@fudan.edu.cn liy@fudan.edu.cn 李弋

A B=A C ∪ ∪ ⇏ B=Ccancellation law 。Example:A={1,2,3},B={3,4,5},C={4,5},

BC,But A B=A C={1,2,3,4,5}∪ ∪Example: A={1,2,3},B={3,4,5},C={3},BC,But A∩B=A∩C={3}A-B=A-C ⇏B=Ccancellation law :symmetric difference

The symmetric difference of A and B, write AB, is the set of all elements that are in A or B, but are not in both A and B, i.e. AB=(A B)-(A∩B)∪ 。

(A B)-(A∩B)=(A-B) (B-A)∪ ∪

)()()()(:Pr BABABABALeftoof

)'()()( lawssMorganDeBABA

)laws vedistributi())(())(( BBAABA

)laws vedistributi()()())()(( BBBAABAA

)())(())(( lawscomplementBAAB

)laws ecommutativ,laws identical()()( ABBA

Theorem 1.4: if AB=AC, then B=C Distributive laws and De Morgan’s laws: B∩(A1∪A2∪…∪An)=(B∩A1)∪(B∩A2)∪…∪(B∩An)

B∪(A1∩A2∩…∩An)=(B∪A1)∩(B∪A2)∩…∩(B∪An)

n

ii

n

ii

n

ii

n

ii AAAA

1111

Chapter 2 Relations

Definition 2.1: An order pair (a,b) is a listing of the objects a and b in a prescribed order, with a appearing first and b appearing second. Two order pairs (a,b) and (c, d) are equal if only if a=c and b=d.

{a,b}={b,a} ，order pairs: (a,b)(b,a) unless a=b.(a,a)

Definition 2.2: The ordered n-tuple (a1,a2,…,an) is the ordered collection

that has a1 as its first element, a2 as its

second element,…, and an as its nth

element.Two ordered n-tuples are equal is only if each corresponding pair of their elements ia equal, i.e. (a1,a2,…,an)=(b1,b2,…,bn) if only if

ai=bi, for i=1,2,…,n.

Definition 2.3: Let A and B be two sets. The Cartesian product of A and B, denoted by A×B, is the set of all ordered pairs ( a,b) where aA and bB. Hence

A×B={(a, b)| aA and bB} Example: Let A={1,2}, B={x,y},C={a,b,c}.A×B={(1,x),(1,y),(2,x),(2,y)};B×A={(x,1),(x,2),(y,1),(y,2)};B×AA×B commutative laws ×

A×C={(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)};

A×A={(1,1),(1,2),(2,1),(2,2)} 。A×=×A=Definition 2.4: Let A1,A2,…An be sets.

The Cartesian product of A1,A2,…An, denoted by A1×A2×…×An, is the set of all ordered n-tuples (a1,a2,…,an) where aiAi for i=1,2,…n. Hence

A1×A2×…×An={(a1,a2,…,an)|aiAi,i=1,2,…,n}.

Example:A×B×C={(1,x,a),(1,x,b),(1,x,c),(1,y,a),

(1,y,b), (1,y,c),(2,x,a),(2,x,b),(2,x,c),(2,y,a),(2,y,b),

(2,y,c)} 。 If Ai=A for i=1,2,…,n, then A1×A2×…×An by An.

Example ： Let A represent the set of all students at an university, and let B represent the set of all course at the university. What is the Cartesian product of A×B?

The Cartesian product of A×B consists of all the ordered pairs of the form (a,b), where a is a student at the university and b is a course offered at the university. The set A×B can be used to represent all possible enrollments of students in courses at the university

students a,b,c, courses:x,y,z,w

(a,y),(a,w),(b,x),(b,y),(b,w) ， (c,w)R={(a,y),(a,w),(b,x),(b,y),(b,w)}RA×B, i.e. R is a subset of A×Brelation

2.2 Binary relations

Definition 2.5: Let A and B be sets. A binary relation from A to B is a subset of A×B. A relation on A is a relation from A to A. If (a,b)R, we say that a is related to b by R, we also write a R b. If (a,b)R , we say that a is not related to b by R, we also write a ℟ b. we say that empty set is an empty relation.

Definition 2.6: Let R be a relation from A to B. The domain of R, denoted by Dom(R), is the set of elements in A that are related to some element in B. The range of R, denoted by Ran(R), is the set of elements in B that are related to some element in A.

Dom(R)A,Ran(R)B 。

Example: A={1,3,5,7},B={0,2,4,6},R={(a,b)|a<b, where aA and bB}Hence R={(1,2),(1,4),(1,6),(3,4),(3,6),

(5,6)}Dom(R)={1,3,5}, Ran(R)={2,4,6}(3,4)R,Because 4<C3, so (4,3)RTable R={(1,2),(1,4),(1,6),(3,4), (3,6),(5,6)}

A B1 21 41 63 43 65 6

A={1,2,3,4},R={(a,b)| 3|(a-b), where a and bA}

R={(1,1),(2,2),(3,3), (4,4),(1,4),(4,1)}Dom R=Ran R=A 。congruence mod 3congruence mod r{(a,b)| r|(a-b) where a and bZ, and

rZ+}

Definition 2.7 ： Let A1,A2,…An be

sets. An n-ary relation on these sets is a subset of A1×A2×…×An.

2.3 Properties of relations

Definition 2.8: A relation R on a set A is reflexive if (a,a)R for all aA. A relation R on a set A is irreflexive if (a,a)R for every aA.

A={1,2,3,4}

R1={(1,1),(2,2),(3,3)} ?

R2={(1,1),(1,2),(2,2),(3,3),(4,4)} ?

Let A be a nonempty set. The empty relation A×A is not reflexive since (a,a) for all aA. However is irreflexive

Definition 2.9: A relation R on a set A is symmetric if whenever a R b, then b R a. A relation R on a set A is asymmetric if whenever a R b, then b℟a. A relation R on a set A is antisymmetric if whenever a R b, then b℟a unless a=b.

If R is antisymmetric, then a ℟ b or b ℟ a when ab.

A={1,2,3,4}S1={(1,2),(2,1),(1,3),(3,1)}?S2={(1,2),(2,1),(1,3)}?S3={(1,2),(2,1),(3,3)} ?

A relation is not symmetric, and is also not antisymmetric

S4={(1,2),(1,3),(2,3)} antisymmetric, asymmetric

S5={(1,1),(1,2),(1,3),(2,3)} antisymmetric, is not

asymmetric

S6={(1,1),(2,2)} antisymmetric, symmetric, is not

asymmetricA relation is symmetric, and is also antisymmetric

Definition 2.10: A relation R on set A is transitive if whenever a R b and b R c, then a R c.

A relation R on set A is not transitive if there exist a,b, and c in A so that a R b and b R c, but a ℟ c. If such a, b, and c do not exist, then R is transitive

T1={(1,2),(1,3)} transitiveT2={(1,1)} transitiveT3={(1,2),(2,3),(1,3)} transitiveT4={(1,2),(2,3),(1,3),(2,1),(1,1)} ?

Example ： Let R be a nonempty relation on a set A. Suppose that R is symmetric and irreflexive. Show that R is not transitive.

Proof: Suppose R is transitive.

Matrix or pictorial represented

Definition 2.11: Let R be a relation from A={a1,a2,…,am} to B={b1,b2,…,bn}. The

relation can be represented by the matrix MR=(mi,j)m×n, where

Rba

Rbam

ji

jiij ),(0

),(1

mi,j=1? ai is related bj

mi,j=0? ai is not related bj

Example: A={1,2,3,4}, R={(1,1),(2,2),(3,3), (4,4),(1,4),(4,1)}, Matrix:

Example ： A={2,3,4},B={1,3,5,7}, <R={(2,3),(2,5), (2,7),(3,5),(3,7),(4,5),(4,7)},Matrix:Let R be a relation on set A. R is reflexive if all the

elements on the main diagonal of MR are equal to 1

R is irreflexive if all the elements on the main diagonal of MR are equal to 0

R is symmetric if MR is a symmetric matrix.

R is antisymmetric if mij=1 with ij, then mji=0

Directed graphs, or Digraphs 。Definition 2.12: Let R be a relation on

A={ a1,a2,…,an}. Draw a small circle (point)

for each element of A and label the circle with the corresponding element of A. These circles are called vertices. Draw an arrow, called an edge, from vertex ai to vertex aj if

only if ai R aj . An edge of the form (a,a) is

represented using an arc from the vertex a back to itself. Such an edge is called a loop.

Example: LetA={1, 2, 3, 4, 5}, R={(1,1),(2,2),(3,3),(4,4), (5,5),(1,4),(4,1),(2,5),(5,2)}, digraph

R1 R∪ 2

R1∩R2

R1-R2

RBAR

2.4 Operations on Relations

1.Inverse relationDefinition 2.13: Let R be a relation from A to

B. The inverse relation of R is a relation from B to A, we write R-1, defined by R-1= {(b,a)|(a,b)R}

Theorem 2.1 ： Let R,R1, and R2 be relation

from A to B. Then(1)(R-1)-1=R;

(2)(R1 R∪ 2)-1=R1

-1 R∪ 2-1;

(3)(R1∩R2)-1=R1

-1∩R2-1;

(4)(A×B)-1=B×A;(5)-1=;

11)6( RR

(7)(R1-R2)-1=R1

-1-R2-1

(8)If R1R2 then R1-1R2

-1

Theorem 2.2 ： Let R be a relation on A. Then R is symmetric if only if R=R-1.

Proof: (1)If R is symmetric, then R=R-1 。RR-1 and R-1R 。(2)If R=R-1, then R is symmetricFor any (a,b)R, (b,a)?R

Exercise: P13 42 43 47 48P126 17,37P134 24, 26,P146 1,2,12, 21,31P167 1,8,9,11