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461191 Discrete Math Lecture 9: Relations

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461191 Discrete Math Lecture 9: Relations San Ratanasanya CS, KMUTNB Adpated from Dr. Goanchanart, RSU – PowerPoint PPT presentation

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Title: 461191 Discrete Math Lecture 9: Relations


1
461191 Discrete Math Lecture 9 Relations
  • San Ratanasanya
  • CS, KMUTNB

Adpated from Dr. Goanchanart, RSU
2
Todays topics
  • Review of Recurrence Relations
  • Administrivia
  • Relations
  • Properties of Relations
  • n-ary Relations
  • Representing Relations
  • Closures of Relations
  • Equivalence Relations
  • Partial Orderings

2
3
Recurrence Relations
  • Recurrence relation ?????? ????? an
    ??????????????? an ???????????????????????????????
    ?????????? ?????? n n0 ??? n0 ???? Nonnegative
    Integer ???????????????????????????????????
    (Solution) ??? relation ??????????
    ???????????????????????????????????????????
  • Relation ???? unique solution ????????????????????
    initial condition
  • ???????????????????????????????????????????
    Recurrence Relation ???

4
Examples
  • ??????????????????????????????? 1 ??? ??? 5 ???
    ???? recurrence relation ?????????????????????????
    ?? ??????????????????????????????????
  • ??? Pn ?????????????????????? n ???
  • P1 1, P2 1, P3 1, P4 1, P5 2
  • P6 3, P7 4, P8 5, .
  • ?????????? P6 3 2 1, P7 4 3 1, P8 5
    4 1,
  • Pn Pn-1 Pn-5

5
Solving Linear Recurrence Relations
  • ????????????????????? Recurrence relation
    ?????????????????????????? recursive ???
    ??????????????????????????????????????????????????
    ????????? n
  • ?????????????????????????? ?????????????????????
    ????????????????????????????????????????? (Linear
    equation)
  • ??????????????????????????????? 2 ?????? ???
  • Homogeneous
  • Non-Homogeneous

6
Examples
  • ????????????????????????????????? an an-1
    2an-2, a0 2, a1 7
  • ??? r2 r 2 0, r1 2, r2 -1
  • ??????? an a12n a2(-1)n
  • ????????? initial condition ?????????????????????
    a1 3 ??? a2 -1
  • ??????????? an 32n -(-1)n

7
Generating Functions
  • ????????????????? (Generating Function)
    ??????????????????????????? Power Series
  • ???????????
  • ?????????? Explicit Formula ??? Recurrence
    Relation
  • ???

8
Examples
  • ??????????????? recurrence relation ak 3ak-1
    ?????? k 1,2,3 ????? initial condition a0 2
  • ??? G(x) ???? generating function ??? ak
    ???????
  • ?????????
  • ?? G(x) ??? xG(x) ????????? recurrence relation
    ????????
  • ???? G(x) 3xG(x) (1-3x)G(x) 2 ??????? G(x)
    2/(1-3x) ???
  • ?????
  • ???????

9
Divide-and-Conquer and Recurrence Relations
  • ???????? Recursive Algorithm ???????????????? n
    ????????????????? ????? a ????????????????????????
    ??????? n/b ?????? n ???b ????????????????????????
    ????????????????????????????? g(n)
    ??????????????????????
  • ????????????????????????????????? Divide
    (???????) ?????????????????????????????
    ??????????????????????????????????????????????
    ??????????????????????????????????????????????????
    ????? ????????????????????????????????????????????
    ??????????????????????????????????????????????????
    Conquer (????????)
  • ???????? f(n) ????????????????????????????????????
    ???????? n ???????? f(n) ???????????????
    Recurrence Relation ??????
  • f(n) af(n/b) g(n)
  • ??????????????????? Divide-and-Conquer Recurrence
    Relation
  • ???????????? Master Theorem ??????????????????????
    ?????? algorithm ????????????

10
Example
  • ??? f(n) f(n/3) 1 ????? n ??? 3 ????? ??? f(1)
    1 ???? f(27)
  • f(27) f(9) 1 f(3) 2 f(1) 3 4

11
Inclusion-Exclusion
12
Example
  • ????????????????????????????????? 200 ????
    Principle of Inclusion-Exclusion
  • 40

13
Administrivia
  • Midterm Exam
  • Statistic Summary
  • n S1S2-MA 20421-2481-3 93
  • Max ??
  • Mean ??
  • Min 6
  • Standard Deviation ??
  • Homework 1-6 and Programming Assignment 1 due
    today
  • Programming Assignment 2 dues next week

Sorry, I cannot finish grading it by today
13
14
Doubts in Midterm Exam
  • 10. ??????????????????????????????????????????????
    1, 2, 3 ???????? 1 ???????? 3
  • ???????? 1 ???????? 3 ??????????? 1 ??? 3
    ?????????? ????? 1 ?????????????
  • ????? 1 ???????? 3 ????????????????? 1
    ?????????????? 3 ????? 1 ????????????????????
  • 16. ????????????????????????????
    ??????????????????????????????? 50 ???
    ????????????????????? 10 ???? ??????????? 85 ????
    ??????????????????????????????????????????????????
    ???????????????????????? ?????????????????????????
    ?????????????? 2 ??????????????????????
  • ???????? ????????????????????? 2
    ????????????????????... ?????????????????????????
    ???????? 2?
  • ????? ?????????? pigeonhole ?N/k? 2 ????? k
    ??????????????? N ???????????????????????????
    ???????? N k 1

15
Doubts in Midterm Exam
  • 18. ??????????.???????? 3 ??? ?????? 2 ??? ???
    ????? 2 ??? ?????.????????????? 1 ???? ??????
    1??? ????????????????????????????????????
  • ???????? ?????????????????????????
    ???????????????????????????????????
  • ????? ???????????????????????? r ?????????????
    ?????????????????????????????????
    ????????????????????????????????????????????
  • 19. ???????????????? CS ?????????????????????
    ????????????????????????????????????????? 40
    ??????????????????????????????? 10 ??
    ??????????????????????????????????????????????????
    ?????
  • ???????? ????????????????????????????????????
    ?????????????????????????
  • ????? ???????????????????????????????????????????
    ?????????? ?????????????????????????????????
    ????????????????????????

16
Relations
  • ????????
  • A ??????????????????
  • B ?????????????????
  • R ????????????????????? (a,b) ???????????
    ???????? a ?????????????????? b
  • ????? ?????????????????? CS01
  • (?????, CS01)
  • ??????? ?????????????????? CS02
  • (???????, CS02)
  • ???????? ?????????????????? CS01
  • (????????, CS01)

17
Relations
  • ????????
  • A ????????????????????????????
  • B ??????????????????
  • R ?????????????????????????????????????? (a,b)
    ????? a ????????? b
  • ???????????? (a,b) ????????? R

18
Relations
  • Relations (????????????) ????????????????
    ??????????????????????????? set
    ??????????????????????????????????????????????????
    ????????? ???? ??????????????????
    ??????????????????????????????????? ???????
  • Binary Relation ??????????????????????? Set 2 set
  • ????? Relation ?????????????????? Binary Relation
  • n-ary Relation ??????????????????? set ??????????
    2 set
  • Ordered Pair ?????????????????????????????????????
    ?? Set 2 set ?????????????????? ????????????
    subset ???????? Cartesian ??????? 2 set ???????
  • ??????????????? a R b ??? relation R ??? a ?? b
    ??? a R b ??????? a ??? b ????? relation R ??????
  • ??????? a R b ??????? (a,b) ?R ???? a ????????
    (related) ??? b ??? R

Definition 1 ??? A ??? B ???? Set
???????? Binary Relation ??? A ?? B ???? subset
??? A x B
Definition 2 Relation ?? Set A ??? Relation ???
A ?? A ??????????? Subset ??? A x A
/
18
19
Examples
  • A 0, 1
  • B a, b
  • A x B
  • (0,a), (0,b), (1,a), (1,b)
  • R1 (0,a), (0,b)
  • R2 (0,a), (1,a)
  • R3 (0,b), (1,a), (1,b)
  • R4 (0,a), (0,b), (1,a), (1,b)
  • ?????????????????????? relation ??? A ????? B

20
Examples
  • ??? A 0, 1, 2 ??? B a, b ???? (0,a),
    (0,b), (1,a), (2,b) ???? relation ??? A ?? B
  • ??? A 1, 2, 3, 4 ???? Ordered pair ?????????

    Relation R (a, b) a divides b
  • R (1, 1), (1, 2), (1, 3), (1, 4), (2, 2),

    (2,4), (3,
    3), (4, 4)

???????????????????? ?????????????????????????
21
Example
  • ?????????????????????????
  • R1 (a,b) a b
  • R2 (a,b) a gt b
  • R3 (a,b) a b or a -b
  • R4 (a,b) a b
  • R5 (a,b) a b 1
  • R6 (a,b) a b 3
  • ????????????????????????????????????
  • (1,1)
  • (1,2)
  • (2,1)
  • (1, -1)
  • (2,2)

22
Properties of Relations
  • R ????????????????? (Reflexive) ??? a R a
    ?????????? a ?? A
  • R ????????????????? (Symmetric) ??? a R b
    ????????? b R a
  • R ???????????????????? (Anti-symmetric) ??? a R b
    ??? b R a ????????? a b
  • R ?????????????????? (Transitive) ??? a R b ??? b
    R c ????????? a R c
  • ?? Definition 3-5 ?? section 8.1

22
23
Examples
  • Relation ?? 1,2,3,4 ?????????? ???? reflexive,
    symmetric, antisymmetric, ??? transitive ????????
    ?
  • R1 (1,1), (1,2), (2,1), (2,2), (3,4), (4,1),
    (4,4)
  • R2 (1,1), (1,2), (2,1)
  • R3 (1,1), (1,2), (1,4), (2,1), (2,2), (3,3),
    (4,1), (4,4)
  • R4 (2,1), (3,1), (3,2), (4,1), (4,2), (4,3)
  • R5 (1,1), (1,2), (1,3), (1,4), (2,2), (2,3),
    (2,4), (3,3), (3,4), (4,4)
  • R6 (3,4)
  • Reflexive R3, R5
  • Symmetric R2, R3
  • Antisymmetric R4, R5, R6
  • Transitive R4, R5, R6

23
24
More on Relations
  • ????????????????????????? Set ????????????????????
    ?? Set Operations ??? Relations ???
  • Functions ??????????????????????????? ???????
    Functions ??????? subset ??? Relations
  • Composite Relation ???? ??????????????????
    ???????????? S?R ?????????????????????????????????
    ???? a R b ??? b S c ??????? a S?R c (??
    Definition 6, Sec. 8.1)
  • Power ??? Relation (Rn) ??????????????????????????
    ??????????????????? R ????? n ?????
    ????????????????????????????? Recursive ?????????
    (?? Definition 7, Sec. 8.1)
  • R1 R, Rn1 Rn?R, n 1, 2, 3,

25
????????????????????????????????
Relation
Function
?
?
One-to-one
?
?
One-to-many
?
?
Many-to-one
26
Examples
  • ??? A 1,2,3 ??? B 1,2,3,4 ????? Relation
    R1 (1,1), (2,2), (3,3) ??? R2 (1,1),
    (1,2), (1,3), (1,4) ????
  • R1 ? R2
  • R1 ? R2
  • R1 - R2
  • R2 R1
  • ???? Composite ??? R ??? S ?????? R ???? relation
    ??? 1,2,3 ????? 1,2,3,4 ?????? R (1,1),
    (1,4), (2,3), (3,1), (3,4) ??? S ???? relation
    ??? 1,2,3,4 ????? 0,1,2 ?????? S (1,0),
    (2,0), (3,1), (3,2), (4,1)
  • S?R (1,0), (1,1), (2,1), (2,2), (3,0), (3,1)

(1,1), (1,2), (1,3), (1,4), (2,2), (3,3)
(1,1)
(2,2), (3,3)
(1,2), (1,3), (1,4)
26
27
Example
  • ??? R (1,1), (2,1), (3,2), (4,3) ???? Rn
  • R2 R?R (1,1), (2,1), (3,1), (4,2)
  • R3 R2?R (1,1), (2,1), (3,1), (4,1)
  • R4 R3?R (1,1), (2,1), (3,1), (4,1)
  • Rn R3, n 3, 4, 5,

28
???????? Relations ?? Set ??????????? n ???
  • ??? set A ???????? n ???.
  • ?????????????? set A ??? subset ??? A x A.
  • ???? A x A ????????????? n2 ???.
  • Set ??????????? m ??????? 2m subset.
  • ??????? A x A ???? subset ??????? 2n2.
  • ???????????? 2n2 relations ?? set ??????????? n
    ???.
  • Example
  • set a, b, c ???????? 232 29 512
    relations.

29
Example
  • ???? Reflexive relation ??????????? set
    ??????????? n ???
  • Reflexive ??? (a, a) ?R ?????????? Product rule
    ???? n(n-1) ordered pair
  • ??????????????? Relations ?????????? 2n(n-1) ???

30
n-ary Relations and its applications
  • ??????????????????????? set ?????????? 2 set
    ?????? ?????????????????????
  • ???? ????????. ??????????????????????? ??. ??????
    ???????????????????, ???????????????
    ???????????????????? ???????? ??????
    ?????????????????????????????? ???????
  • ??? A1, A2, , An ???? set ???? n-ary Relation ??
    set ?????????????? subset ??? A1?A2??An
    ?????? A1, A2, , An ???? Domain ??? Relation ???
    n ?????? degree (?? Definition 1, Sec. 8.2)
  • Example ??? R ???? relation ?? N?N?N ??????????
    triple (a, b, c) ?????? a lt b lt c ??????? (1, 2,
    3)?R ??? (2, 4, 3)?R ?????? Degree ?????? 3 ???
    Domain ??? N

30
31
Database and Relations
  • ??????????????????????????????????????????????????
    ??????????
  • ?????????????????????????????????????????????????
  • ??????????????????????????????????? ????????????
    ?? ????? ?????? ??????????????????????????????????
    ???????????????????????
  • ?????????????????????????????????????????????????
  • ??????????????????????????????????????????????????
    ??????????????????????????? ???? Relational Data
    Model (RDM) ??????????????? IBM ?????????????????
  • ??????????????????????????????????????????????????
    ????? Relational DataBase Management System ????
    RDBMS
  • ???????????????????????????????????????????????
    SQL ???? SEQUEL ???????????? Structured English
    Query Language
  • ????????????????????????????????????? ????
    Oracle, MS Access, mySQL ???????

32
Relational Database
  • ?????????? record (???????) ???????? n-tuples
    ??????????????? fields ???? ???????????? ????
    ??????
  • ???? ?????????. ???? record ??????????????????????
    ?????????. ???? ???? ??????? ??????. ???????? GPA
    ???????
  • ?? RDM ??????????? record ??????????????????????
    n-ary relation
  • ?????????????????????????? ????? (Table)
    ????????????????????????????????????
    ??????????????????????????? Table

33
Relational Database
??????????????????????????????????????????????
  • Primary key ??? Domain (???? Field)
    ???????????????? n-tuple ?? Table ???
  • ???????????? Domain ????????????? Primary key ???
    ????????????????????????? n-ary relation
    ??????????????????????????????????????? ??
    ???????????????????????? ???????? Table
  • Composite key ??? Cartesian product ??? Domain
    ???????????? ?????????? n-tuple ??????????
  • ???? Domain ??? Student_Name ??? Major

34
Operations on n-ary Relations
  • ????????????????????????????????? RDB
    ??????????????????????? ??????????????????????????
    ?????????????????????????????
  • ????????????????? ?????????????????????????
    Condition ??? ???????? ???????????????? n-tuple
    ??????????????? Relation ???? ???? Selection
    Operator ??????? map n-ary ?? Relation ?????
    n-tuple ????????????? Condition ???????
  • ?????????????????????????????????????? ???
    Projection ?????? map ??? n-tuple ?????? m-tuple
  • ???????????? Table ?????????????????? Join
    ???????? Table ???????????????????????????????????
    ????
  • ?? Definiton 2-4 ?? Sec 8.2

35
Operations on n-ary Relations
Definition 2 ??? R ???? n-ary Relation ??? C
???? Condition ??? Element ?? R ??????????
??????? selection operator (Sc) ??????? Map ???
n-ary Relation ?? R ????? n-ary Relation
????????????????? n-Tuple ??? R ????????????
Condition C
Definition 3 Projection Pi1i2,,im ?? Map ???
n-tuple (a1, a2, , an)????? m-tuple (a1, a2, ,
am) ?????? m ? n
Definition 4 ??? R ???? Relation ????? Degree m
??? S ???? Relation ????? Degree n ??? Join
Jp(R, S) ?????? p ? m ??? p ? n ???? Relation
????? Degree m n - p ??????????????????? (m n
p)-tuple (a1, a2, , am-p, c1, c2, , cp, b1,
b2, , bn-p) ?????? m-tuple (a1, a2, , am-p, c1,
c2, , cp) ????? R ??? n-tuple (c1, c2, , cp,
b1, b2, , bn-p) ????? S
36
Examples
  • ????????? C1 (Major Computer Science ? GPA
    gt 3.5) ???????????????? n-ary Relation ?? Table 1
    ????????????????????
  • ??? 4-tuple ??????????? (Ackermann, 231455,
    Computer Science, 3.88)
  • ?????? Projection P1,3 ??? 4-tuples (2,3,0,4),
    (Jane Doe, 234111001, Geogrpahy, 3.14), (a1, a2,
    a3, a4) ????????????????????????
  • ??? (2,0), (23411101, Geography), (a1, a3)
  • ??????????????????? Projection P1,4 ??? Table 1
    ???????
  • ??????????? Join Table 5 ??? 6 ???????????????????
    ????

37
Example
38
SQL Examples
SELECT Departure_Time FROM Flights WHERE
Destination Detroit
  • Select ?? SQL ??????? Projection
  • ????????????? Database ??? SQL ????????????? ??.
    ????? ??????????? Database

SELECT Professor, Time FROM Teaching_assignments,
Class_schedule WHERE Department Mathematics
(Rosen, 300 P.M.)
39
??????????????????? (Representing Relations)
  • ?????????????????????????????????????? ????
    Ordered pair ???????????????????????????????
  • ??????????? ????????????????????????????????????
    Zero-One Martix ??? Directed Graph
    ?????????????????????
  • ??????????? Zero-One Matrix ??????????????????????
    ?????????????????????????
  • Directed Graph ???????????????????????????????????
    ?????????????????????????????????????

39
40
Representing Relations Using Matrices
  • Relation ??????? Finite Set ?????????????????
    Zero-One Matrix ????????? R ???? Relation ??? A
    ????? B ?????? Element ?? A ???B
    ?????????????????????????? ???????????????????????
    ??? ??????? Relation R ????????????????? Matrix
    MR mij ??????
  • ??????? ??? Element ?? Matrix ??? Row ???
    Column ij ??????????? ????? ??? ai
    ????????????????? bj ?????????? R
    ?????????????????????????? ?????
  • Example ????? A 1,2,3, B 1,2 ??? R ????
    relation ??? A ?? B ?????? R (a,b) a?A ? b?B
    ? a gt b ???? Matrix ??????? R ???????? ?????? a1
    1, a2 2, a3 3, b1 1, b2 2
  • ???????? R (2,1), (3,1), (3,2) ???????

40
41
Representing Relations Using Matrices
  • Matrix ??????? Relation ?? Set ???????????????????
    ???????? Relation ?????????
  • Relation R ?? A ?????? Reflexive ?????????? mii
    1 ???????? ?????????????????????????? ??? Matrix
    ??????? R ???? ????????? 1 ??????
  • Relation R ?? A ?????? Symmetric ?????????? mij
    mji ???????? ????????? (i,j) ??? (j,i) ??? Matrix
    ??????? R ???? ?????????????????? ???? MR MRT
  • Relation R ?? A ?????? Anti-symmetric ??? mij 1
    ???? mij 0 ????? i ? j ???? ??? mij 0 ????
    mij 0 ????? i ? j

42
Representing Relations Using Matrices
  • Boolean Operation ??? Matrix ???????????????
    Union ??? Intersect ??? Relation ???
  • ?????????? ??????????? Composite of Relation
    ?????? Matrix ?????????? ????????? Boolean Product

43
Examples
  • ??? Relation R ?? Set ???????? Matrix ????????
    ??????? R???? Reflexive, Symmetric ???/????
    Antisymmetric
  • ????????? Relation R1 ??? R2 ?? Set A ????????
    Matrix ???????? ???? Matrix ??????? R1 ? R2 ???
    R1 ? R2

R ?????? Reflexive ??? Symmetric ??????????
Antisymmetric
44
Examples
  • ???? Matrix ??????? Relation SoR ????? Matrix
    ???? R??? S ???
  • ???? Matrix ??????? R2 ???????????

45
Representing Relations Using Directed Graphs
  • ????????????????????? Relation ???????????????
    ??????????? Graph ??????????? Element ??? Set
    ?????????????? ???????? Ordered Pair
    ?????????????????????????????????
    ?????????????????? ????????????????????????????
    Directed Graph ???? Digraph

Definition 1 Directed Graph ???? Digraph
???????????? Set V ??? Vertices ???? Nodes
????????? Set E ??? Ordered Pair ??? Element ??
V ??????????? Edges ???? Arcs ?????? Vertex a
?????????? Initial Vertex ??? Edge (a, b) ???
Vertex b ?????????? Terminal Vertex ??? Edge ???
Edge ???????? (a, a) ???????????????????????????
????? Vertex a ??????????????? ???? Edge
??????????? ??????????? Loop
45
46
Representing Relations Using Digraphs
  • ???????????? Directed Graph ??????????????????????
    ?????????? Relation ??? ????
  • ??? Relation ???? Reflexive ???? ???????? Loop
    ?????? Vertex ??? Directed Graph ????
  • ??? Relation ???? Symmetric ???? ??????? Edge
    ???????????????? Vertex ???????????????
    ??????????? Edge ??????????????????????????
  • ??? Relation ???? Antisymmetric ??????????? Edge
    ???(?????????) ???????? Vertex ??????????
  • ??? Relation ???? Transitive ??????????? ?????
    Edge ??? Vertex x ????? y ?????? Vertex y ????? z
    ???? ???????? Edge ??? Vertex x ????? z ????

46
47
Examples
  • ????? Directed Graph ????????????? Vertex a, b,
    c, ??? d ??? Edge (a, b), (a, d), (b, b), (b,
    d), (c, a), (c, b), ??? (d, b)
  • ???? Ordered Pair ?? Relation R ???????????
    Directed Graph ????????

48
Example
  • ??????? Relation ??????????? Directed Graph
    ?????????????????? Reflexive, Symmetric,
    Antisymmetric ???/???? Transitive

Symmetric
49
Closures of Relations
  • ????????????????? Network, ???????????????????????
    ??????????????? (Node) ???????????
    ??????????????????????????????????????? Data link
    ???????????????????? ??????????????
    ??????????????????????????????????????????????????
    ????? ???????????????????? Node
    ?????????????????????????????????????????? Node
    ???????????????? ????????????????? link
    ??????????????????????????????????????????????????
    ???????????? Relation ???????????????????????????
    ????????????????????? transitive closure ????
    ??????????????????????????

49
50
Closures of Relations
  • ?????? R???? Relation ?? Set A, Relation
    R???????????????????????????? P ????????????
    Reflexive, Symmetric ???? Transitive
    ?????????????? Relation ?????????????? P ????????
    ??? S???????????? R ?????? S ?????? Subset
    ??????? Relation ?????????????? P ????? R
    ?????????? ????????????????? S ??????? Closure
    ??? R ????????????? P

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Reflexive Closure
  • ??????? Relation R (1,1), (1,2), (2,1), (3,2)
    ?? Set A 1, 2, 3 ????????? R ???????
    Reflexive ?????????? Reflexive Relation
    ??????????????????? R ?????????????????????????
  • ????? (2,2) ??? (3,3) ???? R ??????? R ????
    Reflexive
  • ???????????? Ordered pair ??? (a, a)
    ????????????????? R
  • ??????????????? Relation ????????? R ??????????
    ????????????? Reflexive Relation ??????????? R
    ?????????? ???????????????? (2,2) ??? (3,3)
  • ??????? ????????? Relation R ????? (2,2) ???
    (3,3) ?????? ???? Reflexive ???????
    ???????????????? Reflexive Relation ????? R
    ?????? ???????????????? Reflexive Closure ??? R
  • ???????? Relation R ?? Set A ??????????????
    Reflexive Closure ??? R?????????????????? ??????
    (a,a), a ?A ????????????????? R?????? R
  • ????????????????? Reflexive Closure ??? R
    ??????????? R???????? ? ??? Diagonal Relation ??
    A ??? ? (a,a) a ?A

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Symmetric Closure
  • ??????? Relation R (1,1), (1,2), (2,2), (2,3),
    (3, 1), (3,2) ?? Set A 1, 2, 3 ????????? R
    ??????? Symmetric ?????????? Symmetric Relation
    ??????????????????? R ?????????????????????????
  • ????? (2,1) ??? (1,3) ???? R ??????? R ????
    Symmetric
  • ???????????? Ordered pair ??? (b, a) ??? (a, b) ?
    R ?????????????? R
  • ??????????????? Relation ????????????????? R
    ?????????????? Symmetric Relation ???????????? R
    ?????????? ???????????????? (2,1) ??? (1,3)
  • ??????? ????????? Relation R ????? (2,1) ???
    (1,3) ?????? ???? Symmetric ???????
    ???????????????? Symmetric Relation ????? R
    ?????? ???????????????? Symmetric Closure ??? R
  • ???????? Relation R ?? Set A ??????????????
    Symmetric Closure ??? R????????? union R ????
    Inverse ??? R ???? R-1
  • ????????????????? Symmetric Closure ??? R
    ??????????? R? R-1 ?????? R-1 (b,a) (a,b)
    ?R

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Examples
  • ???? Reflexive Closure ??? Relation R (a,b)
    a lt b ?? set ??? Integer
  • ???? Symmetric Closure ??? Relation R (a,b)
    a gt b ?? set ??? Positive Integer

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Transitive Closure
  • ???????? Relation R (1,3), (1,4), (2,1),
    (3,2) ?? Set A 1, 2, 3, 4 ????????? R
    ??????? Transitive ?????????? Transitive Relation
    ??????????????????? R ?????????????????????????
  • ????????????? (1,2), (2,3), (2,4) ??? (3,1) ????
    R ????????????????? R ???? Transitive ???
    ???????????????????????? (3.4)
  • ????????????????? ordered pair ????????????
    ?????????????????????????? R ???? Transitive
  • ??????????????? Transitive Closure
    ???????????????????????? Reflexive ??? Symmetric
  • ?????????? Transitive Closure????
    ??????????????????????????????????? ???? Path
    ????

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Paths in Directed Graphs
  • ????????? Transitive Relation ???? Digraph
    ????????????????????? path ????

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Example
  • ???????????????????????? Path ??? Directed Graph
    ????????????????????

a,b,d,e ???? Path ???????????? 3
a,e,c,d,b ??????? Path ????????? (c,d) ?????? Edge
b,a,c,b,a,a,b ???? Path ???????????? 6
d,c ???? Path ???????????? 1
c,b,a, ???? Path ???????????? 2
e,b,a,b,a,b,e ???? Path ???????????? 6
????????????? ???? 2 Path ??????? Circuit ???
b,a,c,b,a,a,b ??? e,b,a,b,a,b,e
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Paths in Directed Graphs
  • ??????? Path ???????? Relation ???????????
    Directed Graph ??????????? ???? Path ??? a ?? b
    ?? R ????? Sequence ??? Element a, x1, x2,,
    xn-1,b ?????? (a, x1) ? R, (x1, x2) ? R, ,
    (xn-1,b) ? R ??????????? Theorem 1
  • ??????????? Path ????????????????? Transitive ???
    Relation ????????????????????????? Vertices ??
    Directed Graph ?????? Path ?????????????????

Theorem 1 ??? R ???? Relation ?? Set A ???? Path
???????????? n ?????? n ???? Positive Integer ???
a ?? b ?????????? (a,b) ? Rn
Definiton 2 ??? R???? Relation ?? Set ,
Connectivity Relation R ????????????????????
(a,b) ????? Path ????????????????????? ????? ???
a?? b ?? R
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Transitive Closure
  • ????????? Rn ?????????????????? (a,b) ???????
    path ??????? n ??? a ?? b ????? R ???? union ???
    Rn ??????? ????
  • ??? Theorem 2 ????????? ????? Transitive Closure
    ???????? Connectivity Relation ???????
  • Example ??? R ???? Relation ?? Set
    ?????????????? ????????????????? (a,b) ??? a
    ?????? b ???? Rn ??? R ?????? n ???? positive
    integer ?????????? 1

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Transitive Closure
  • ??????????????????? R ?????????????????????? R
    ??? path ?????????????????? ??????????????????????
    ?? path ?????????????????? R ????????? Lemma 1
  • ??????? Lemma 1 ?????
  • ??????????????? Matrix ??????? R ???
    (?????????????? Defiiniton 3 ??? Algorithm 1)
    ???????????? O(n4) (?????????????? Sec. 8.4)
  • Roy Warshall ??????? algorithm ????????????? R
    ???????????????????????????? Matrix ?????????????
    O(n3)

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Warshalls Algorithm
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Example
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Washalls Algorithm (Cont.)
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Warshalls Algorithm
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Equivalence Relations
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Equivalence Relations
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Equivalence Class Partition
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Partial Orderings
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Lexicographic Ordering
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Hasse Diagram
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Minimal and Maximal
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The Greatest/The Least Element
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Upper Bound and Lower Bound
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LUB and GLB
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Lattice
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Topological Sorting
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Homework 8
  • Section 8.1
  • 8, 9, 13, 24, 27, 33
  • Section 8.2
  • 10, 26, 19, 28, 29
  • Section 8.3
  • 31, 32
  • Section 8.4
  • 15, 35
  • Section 8.5
  • 9, 11, 15, 16
  • Section 8.6
  • 12, 13, 40, 64, 65
  • Supplementary
  • ---

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