Grimaldi, Discrete and Combinatorial Mathematics 5/e, Addison-Wesley

29 Pages·2017·2.66 MB·English

Computer Science and Information Engineering
National Chi Nan University
Combinatorial Mathematics
Dr. Justie Su-Tzu Juan
Chapter 7 Relations: The Second
Time Around
§ 7.1 Relations Revisited: Properties of
Relations
Slides for a Course Based on the Text
Discrete & Combinatorial Mathematics (5th Edition)
by Ralph P. Grimaldi
(c) Spring 2021, Justie Su-Tzu Juan
§ 7.1 Relations Revisited: Properties of Relations
Def 7.1 : A, B: sets, R  A  B: R is called a relation from A to B;
R  A  A: R is called a relation on A;
Ex 7.1 :
Z
a) Defined R on by a R b or (a, b)  R, if a  b:
Z
R is the ordinary “less than or equal to” relation on .
Z Q R C
( 可改成 , , but not on )
Z Z
b) Let n  +, Define R on by x R y, if n(x – y):
Z
R is the modulo n relation on .
ex. n = 7: 9R 2, –3R 11, (14, 0)  R, 3R 7 (3 is not related to 7).
c) Let U = {1, 2, 3, 4, 5, 6, 7}, C  U, C = {1, 2, 3, 6}
Define R on P(U) by A R B, if A  C = B  C
ex: {1, 2, 4, 5} and {1, 2, 5, 7} are related,
X = {4, 5} and Y = {7} are related;
S = {1, 2, 3, 4, 5} and T= {1, 2, 3, 6, 7} are not related (S R T)
(c) Spring 2021, Justie Su-Tzu Juan
§ 7.1 Relations Revisited: Properties of Relations
Def 7.2 : A relation R on A is called reflexive 
 x  A, (x, x)  R.
Ex 7.5 : If A= n, A  A= n2, there are 2n2 relations on A.
How many of these are reflexive?
2(n2–n)
Def 7.3 : A relation R on A is called symmetric
  x, y  A, (x, y)  R  (y, x)  R .
Note : Let A= n
1) How many relations on A are symmetric? 2(n2+n)/2
2) Both reflexive and symmetric? 2(n2–n)/2
(c) Spring 2021, Justie Su-Tzu Juan
§ 7.1 Relations Revisited: Properties of Relations
Def 7.4 : A relation R on A is called transitive
  x, y, z  A, (x, y), (y, z)  R  (x, z)  R.
Def 7.5 : A relation R on A is called antisymmetric
  a, b  A, (a R b and b R a)  a = b.
Note : How many relations of A are antisymmetric? (|A| = n) ?
(2n)(3(n2–n)/2)
1) (x, x)  A  A (n : 2n)
2) (x, y)  A  A, x  y: (a) (y, x)  R , (b) (x, y)  R ,
(c) (x, y), (y, x)  R
( (n2–n)/2 : 3(n2–n)/2)
(c) Spring 2021, Justie Su-Tzu Juan
§ 7.1 Relations Revisited: Properties of Relations
Def 7.6 : A relation R on A is called a partial order or a partial
ordering relation, if R is reflexive, antisymmetric, and
transitive.
Ex 7.14 : EX 7.1(a) is a partial order,
EX 7.1(b) is not (∵ not antisymmetric),
Def 7.7 : An equivalence relation R on a set A is a relation that
is reflexive, symmetric, and transitive.
Ex 7.16 (1/2):
a) Ex 7.1(b) is equivalence relation.
(c) Spring 2021, Justie Su-Tzu Juan
§ 7.1 Relations Revisited: Properties of Relations
b) Let A = {1, 2, 3}, then
R = {(1, 1), (2, 2), (3, 3)}
1
R = {(1, 1), (2, 2), (2, 3), (3, 2), (3, 3)}
2
R = {(1, 1), (1, 3), (2, 2), (3, 1), (3, 3)}
3
R = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
4
are all equivalence relations on A?
c) For a given finite set A = {a , a , …, a },
1 2 n
A  A : the largest equivalence relation on A.
R = {(a , a ) | 1  i  n} : the smallest equivalence relation on A.
i i
(equality relation)
e) If R is a relation on A, then R is both an equivalent relation
and a partial order on A if and only if R is the equality
relation on A.
(c) Spring 2021, Justie Su-Tzu Juan
Computer Science and Information Engineering
National Chi Nan University
Discrete Mathematics
Dr. Justie Su-Tzu Juan
Chapter 7 Relations: The Second
Time Around
§ 7.2 Computer Recognition: Zero-One
Matrices and Directed Graphs (1)
Slides for a Course Based on the Text
Discrete & Combinatorial Mathematics (5th Edition)
by Ralph P. Grimaldi
(c) Spring 2021, Justie Su-Tzu Juan
§ 7.2 Computer Recognition: Zero-One Matrices and
Directed Graphs
Def 7.8 : A, B, C: sets, R  A  B, R  B  C. The
1 2
composite relation R 。R  A  C defined by
1 2
R 。R = {(x, z) | x  A, z  C, and  y  B with (x, y)
1 2
 R , (y, z)  R }.
1 2
Ex 7.17: A = {1, 2, 3, 4}, B = {w, x, y, z}, C = {5, 6, 7}
back
R = {(1, x), (2, x), (3, y), (3, z)}  A  B
1
R = {(w, 5), (x, 6)}  B  C
2
R = {(w, 5), (w, 6)}  B  C
3
R 。R ={(1, 6), (2, 6)} R 。R = 
1 2 1 3
(c) Spring 2021, Justie Su-Tzu Juan
§ 7.2 Computer Recognition: Zero-One Matrices and
Directed Graphs
Ex 7.18 : A: employees, B: programming languages,
C = {p , p , …, p }: projects.
1 2 8
R  A  B : (x, y)  R means x is proficient in y,
1 1
R  B  C : (y, z)  R means z need y.
2 2
 R 。R has been used to set up a matching
1 2
process between employees and projects on the
basis of employee knowledge of specific
programming languages.
(c) Spring 2021, Justie Su-Tzu Juan
§ 7.2 Computer Recognition: Zero-One Matrices and
Directed Graphs
Thm 7.1 : A, B, C, D : sets, R  A  B, R  B  C, R  C  D.
1 2 3
The R 。(R 。R ) = (R 。R )。R .
1 2 3 1 2 3
Proof.
1. R 。(R 。R )  A  D, (R 。R )。R  A  D.
1 2 3 1 2 3
2.  (a, d)  R 。(R 。R )
1 2 3
  b  B s.t. (a, b)  R  (b, d)  R 。R
1 2 3
  c  C s.t. (b, c)  R  (c, d)  R
2 3
∵ (a, b)  R  (b, c)  R  (a, c)  R 。R
1 2 1 2
∵ (a, c)  R 。R  (c, d)  R
1 2 3
 (a, d)  (R 。R )。R
1 2 3
∴ R 。(R 。R )  (R 。R )。R
1 2 3 1 2 3
Similar, (R 。R )。R  R 。(R 。R )
1 2 3 1 2 3
R 。(R 。R ) = (R 。R )。R
1 2 3 1 2 3
(c) Spring 2021, Justie Su-Tzu Juan