chapter 3 fuzzy relation and composition. 3.1 crisp relation 3.1.1 product set definition (product...
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CHAPTER 3 FUZZY RELATION and COMPOSITION
3.1 Crisp relation
3.1.1 Product set
Definition (Product set) Let A and B be two non-empty sets, the product set or Cartesian product A B is defined as follows,
A B {(a, b) | a A, b B }
extension to n sets
A1A2...An(= )={(a1, ... , an) | a1 A1, a2 A2, ... , an An }
n
iiA
1
3.1 Crisp relation
Example 3.1 A {a1, a2, a3}, B {b1, b2}
A B {(a1, b1), (a1, b2), (a2, b1), (a2, b2), (a3, b1), (a3, b2)}
(a1, b2) (a2, b2)
(a1, b1)
(a3, b2)
(a2, b1) (a3, b1)
b2
a1
b1
a3a2
Fig 3.1 Product set A B
3.1 Crisp relation
A A {(a1, a1), (a1, a2), (a1, a3), (a2, a1), (a2, a2), (a2, a3),
(a3, a1), (a3, a2), (a3, a3)}
(a1, a2) (a2, a2)
(a1, a1)
(a3, a2)
(a2, a1) (a3, a1)
a2
a1
a1
a3a2
(a1, a3) (a2, a3) (a3, a3) a3
Fig 3.2 Cartesian product A A
3.1 Crisp relation
3.1.2 Definition of relation Binary Relation
If A and B are two sets and there is a specific property between elements x of A and y of B, this property can be described using the ordered pair (x, y). A set of such (x, y) pairs, x A and y B, is called a relation R.
R = { (x,y) | x A, y B } n-ary relation
For sets A1, A2, A3, ..., An, the relation among elements x1 A1, x2 A2, x3
A3, ..., xn An can be described by n-tuple (x1, x2, ..., xn). A collection o
f such n-tuples (x1, x2, x3, ..., xn) is a relation R among A1, A2, A3, ..., An.
(x1, x2, x3, … , xn) R , R A1 A2 A3 … An
3.1 Crisp relation
Domain and Rangedom(R) = { x | x A, (x, y) R for some y B }
ran(R) = { y | y B, (x, y) R for some x A }
Rdom(R) ran(R)
A B
fx1
x2
x3
A B
y1
y2
y3
Mapping y f(x) dom(R) , ran(R)
3.1 Crisp relation
3.1.3 Characteristics of relation(1) One-to-many
x A, y1, y2 B (x, y1) R, (x, y2) R
(2) Surjection (many-to-one) f(A) B or ran(R) B. y B, x A, y f(x) Thus, even if x1 x2, f(x1) f(x2) can hold.
A B
y
1xy
2
One-to-many relation(not a function)
fx1
x2
A B
y
Surjection
3.1 Crisp relation
(3) Injection (into, one-to-one) for all x1, x2 A, x1 x2 , f(x1) f(x2). if R is an injection, (x1, y) R and (x2, y) R then x1 x2.
(4) Bijection (one-to-one correspondence) both a surjection and an injection.
fx1
A B
y1x2 y2x3 y3
y4
Injection
fx1
x2
A B
y1
y2x3 y3x4 y4
Bijection
3.1 Crisp relation
3.1.4 Representation methods of relations (1)Bipartigraph(Fig 3.7)
representing the relation by drawing arcs or edges
(2)Coordinate diagram(Fig 3.8)
plotting members of A on x axis and that of B on y axis
Fig 3.8 Relation of x2 + y2 4
A B
a1 b1
a2
b2
a3
b3a4
Fig 3.7 Binary relation from A to B
4
-4
-4 4
y
x
3.1 Crisp relation
(3) Matrix
manipulating relation matrix
(4) Digraph(Fig 3.9)
the directed graph or digraph method
R b1 b2 b3
a1 1 0 0
a2 0 1 0
a3 0 1 0
a4 0 0 1
MR (mij)
Rba
Rbam
ji
jiij ),(,0
),(,1
i 1, 2, 3, …, mj 1, 2, 3, …, n
2 3
4
1
Matrix Fig 3.9 Directed graph
3.1 Crisp relation
3.1.5 Operations on relations R, S A B
(1) Union of relation T R S
If (x, y) R or (x, y) S, then (x, y) T
(2) Intersection of relation T R S
If (x, y) R and (x, y) S, then (x, y) T.
(3) Complement of relation
If (x, y) R, then (x, y)
(4) Inverse relation
R-1 {(y, x) B A | (x, y) R, x A, y B}
(5) Composition T
R A B, S B C , T S R A C
T {(x, z) | x A, y B, z C, (x, y) R, (y, z) S}
R
3.1 Crisp relation
3.1.6 Path and connectivity in graph Path of length n in the graph defined by a relation R A A is a
finite series of p a, x1, x2, ..., xn-1, b where each element should be
a R x1, x1 R x2, ..., xn-1 R b.
Besides, when n refers to a positive integer
(1) Relation Rn on A is defined, x Rn y means there exists a path from x to y whose length is n.
(2) Relation R on A is defined, x R y means there exists a path from x to y. That is, there exists x R y or x R2 y or x R3 y ... and . This relation R is the reachability relation, and denoted as xRy
(3) The reachability relation R can be interpreted as connectivity relation of A.
3.2 Properties of relation on a single set
3.2.1 Fundamental properties 1) Reflexive relation
x A (x, x) R or R(x, x) = 1, x A
irreflexive if it is not satisfied for some x A
antireflexive if it is not satisfied for all x A
2) Symmetric relation (x, y) R (y, x) R or R(x, y) = R(y, x), x, y A
asymmetric or nonsymmetricwhen for some x, y A, (x, y) R and (y, x) R.
antisymmetricif for all x, y A, (x, y) R and (y, x) R
3.2 Properties of relation on a single set
3) Transitive relation For all x, y, z A
(x, y) R, (y, z) R (x, z) R
4) Closure The requisites for closure
(1) Set A should satisfy a certain specific property.
(2) Intersection between A's subsets should satisfy the relation R. Closure of R
the smallest relation R' containing the specific property
3.2 Properties of relation on a single set
Example 3.3 The transitive closure (or reachability relation) R of R
for A {1, 2, 3, 4} and R {(1, 2), (2, 3), (3, 4), (2, 1)} is
R R R2 R3 …
={(1, 1), (1, 2), (1, 3), (1,4), (2,1), (2,2), (2, 3), (2, 4), (3, 4)}.
1 3
4
2
(a) R
1 3
4
2
(b) R
Fig 3.10 Transitive closure
3.2 Properties of relation on a single set
3.2.2 Equivalence relation
(1) Reflexive relation
x A (x, x) R
(2) Symmetric relation
(x, y) R (y, x) R
(3) Transitive relation
(x, y) R, (y, z) R (x, z) R
3.2 Properties of relation on a single set
Equivalence classes
a partition of A into n disjoint subsets A1, A2, ... , An
(a) Expression by set
b
ad
c
e
(b) Expression by undirected graph
Fig 3.11 Partition by equivalence relation(A/R) {A1, A2} {{a, b, c}, {d, e}}
b
a
c
d e
A1 A2
A
3.2 Properties of relation on a single set
3.2.3 Compatibility relation (tolerance relation) (1) Reflexive relation (2) Symmetric relation
x A (x, x) R (x, y) R (y, x) R
(a) Expression by set
b
ad
c
e
(b) Expression by undirected graph
Fig 3.12 Partition by compatibility relation
b
a
c
d e
A1 A2
A
3.2 Properties of relation on a single set
3.2.4 Pre-order relation (1) Reflexive relation
x A (x, x) R
(2) Transitive relation
(x, y) R, (y, z) R (x, z) R
h g
fd
b
c
e
a
A
(a) Pre-order relation
a
c
b, d
e
f, h g
(b) Pre-order
Fig 3.13 Pre-order relation
3.2 Properties of relation on a single set
3.2.5 Order relation (1) Reflexive relation
x A (x, x) R
(2) Antisymmetric relation(x, y) R (y, x) R
(3) Transitive relation (x, y) R, (y, z) R (x, z) R
strict order relation(1’) Antireflexive relation
x A (x, x) R total order or linear order relation
(4) x, y A, (x, y) R or (y, x) R
3.2 Properties of relation on a single set
Ordinal function For all x, y A (x y),
1) If (x, y) R, xRy or x > y,f(x) f(y) + 1
2) If reachability relation exists in x and y, i.e. if xRy, f(x) > f(y)
b
a
d
c e
f
g
(a) Order relation a
d
eb
c f g1
2
3
(b) Ordinal function Fig 3.14 Order relation and ordinal function
3.2 Properties of relation on a single set
Table 3.1 Comparison of relations
Property
RelationReflexive Antireflexive Symmetric Antisymmetric Transitive
Equivalence Compatibility
Pre-order Order
Strict order
3.3 Fuzzy relation
3.3.1 Definition of fuzzy relation Crisp relation
membership function R(x, y)
R : A B {0, 1}
Fuzzy relation
R : A B [0, 1]
R = {((x, y), R(x, y))| R(x, y) 0 , x A, y B}
R (x, y) = 1 iff (x, y) R
0 iff (x, y) R
3.3 Fuzzy relation
(a1, b1) ...(a1, b2 ) ( a2, b1)
0.5
1
R
BAR
BA
Fig 3.15 Fuzzy relation as a fuzzy set
3.3 Fuzzy relation
3.3.2 Examples of fuzzy relation Example 3.3
Crisp relation R
R(a, c) 1, R(b, a) 1, R(c, b) 1 and R(c, d) 1.
Fuzzy relation R
R(a, c) 0.8, R(b, a) 1.0, R(c, b) 0.9, R(c, d) 1.0 a
b
c
d
a
b
c
d
0.8
1.0
0.9 1.0
(a) Crisp relation (b) Fuzzy relation
Fig 3.16 crisp and fuzzy relations
A A
a b c d
a 0.0 0.0 0.8 0.0
b 1.0 0.0 0.0 0.0
c 0.0 0.9 0.0 1.0
d 0.0 0.0 0.0 0.0
corresponding fuzzy matrix
3.3 Fuzzy relation
3.3.3 Fuzzy matrix
1) Sum
A + B Max [aij, bij]
2) Max product
A B AB Max [ Min (aik, bkj) ]
k
3) Scalar product
A where 0 1
3.3 Fuzzy relation
Example 3.4
3.3 Fuzzy relation
3.3.4 Operation of fuzzy relation 1) Union relation
(x, y) A B
R S (x, y) Max [R (x, y), S (x, y)] R (x, y) S (x, y)
2) Intersection relation
R S (x) = Min [R (x, y), S (x, y)] = R (x, y) S (x, y)
3) Complement relation
(x, y) A B
R (x, y) 1 - R (x, y)
4) Inverse relation
For all (x, y) A B, R-1 (y, x) R (x, y)
3.3 Fuzzy relation
Fuzzy relation matrix
3.3 Fuzzy relation
3.3.5 Composition of fuzzy relation For (x, y) A B, (y, z) B C,
SR (x, z) = Max [Min (R (x, y), S (y, z))]
y
= [R (x, y) S (y, z)] y
MS R MR MS
3.3 Fuzzy relation
Example 3.6
=>
=>
Composition of fuzzy relation
3.3 Fuzzy relation
3.3.6 -cut of fuzzy relation R = {(x, y) | R(x, y) , x A, y B} : a crisp relation.
Example 3.7
3.3 Fuzzy relation
3.3.7 Projection and cylindrical extensionfor (x, y) A B
is a value in the level set; R is a -cut relation; R is a fuzzy relation
Example 3.8
yxyx RR ,,
3.3 Fuzzy relation
ProjectionFor all x A, y B,
: projection to A
: projection to B
Example 3.9
yxMaxx Ry
RA,
yxMaxy Rx
RB,
3.3 Fuzzy relation
Projection in n dimension
Cylindrical extension
C(R) (a, b, c) R (a, b)
a A, b B, c C
Example 3.10
nRXXX
ikiiR xxxxxx Maxjmjj
ikXiXiX,,,,,, 21
,,,21
2121
3.4 Extension of fuzzy set
3.4.1 Extension by relation Extension of fuzzy set
x A, y B y f(x) or x f -1(y)
for y B if f -1(y)
Example 3.11 A {(a1, 0.4), (a2, 0.5), (a3, 0.9), (a4, 0.6)}, B {b1, b2, b3}
f -1(b1) {(a1, 0.4), (a3, 0.9)}, Max [0.4, 0.9] 0.9
B' (b1) 0.9
f -1(b2) {(a2, 0.5), (a4, 0.6)}, Max [0.5, 0.6] 0.6
B' (b2) 0.6
f -1(b3) {(a4, 0.6)}
B' (b3) 0.6
B' {(b1, 0.9), (b2, 0.6), (b3, 0.6)}
xy Ayfx
B Max 1
3.4.2 Extension principle Extension principle
A 1 A2 ... Ar ( x1 x2 ... xr ) Min [ A1 (x1), ... , Ar(xr) ]
f(x1, x2, ... , xr) : X Y
otherwise , ,,
if , 0
1,,,
1
1
21
rAAxxxfy
B
xxMinMax
yf
y
r
r
3.4 Extension of fuzzy set
3.4.3 Extension by fuzzy relation For x A, y B, and B B
B' (y) Max [Min (A (x), R (x, y))]
x f -1(y)
Example 3.12 For b1 Min [A (a1), R (a1, b1)] Min [0.4, 0.8] 0.4
Min [A (a3), R (a3, b1)] Min [0.9, 0.3] 0.3
Max [0.4, 0.3] 0.4 B' (b1) 0.4
For b2, Min [A (a2), R (a2, b2)] Min [0.5, 0.2] 0.2
Min [A (a4), R (a4, b2)] Min [0.6, 0.7] 0.6
Max [0.2, 0.6] 0.6 B' (b2) 0.6
For b3, Max Min [A (a4), R (a4, b3)]
Max Min [0.6, 0.4] 0.4 B' (b3) 0.4
B' {(b1, 0.4), (b2, 0.6), (b3, 0.4)}
3.4 Extension of fuzzy set
Example 3.13 A {(a1, 0.8), (a2, 0.3)}
B {b1, b2, b3}
C {c1, c2, c3}
B' {(b1, 0.3), (b2, 0.8), (b3, 0)}
C' {(c1, 0.3), (c2, 0.3), (c3, 0.8)}
3.4 Extension of fuzzy set
3.4.4 Fuzzy distance between fuzzy sets Pseudo-metric distance
(1) d(x, x) 0, x X
(2) d(x1, x2) d(x2, x1), x1, x2 X
(3) d(x1, x3) d(x1, x2) d(x2, x3), x1, x2, x3 X
+ (4) if d(x1, x2)=0, then x1 = x2 metric distance
Distance between fuzzy sets
, d(A, B)() Max [Min (A(a), B(b))]
d(a, b)
3.4 Extension of fuzzy set
Example 3.14 A {(1, 0.5), (2, 1), (3, 0.3)} B {(2, 0.4), (3, 0.4), (4, 1)}.
3.4 Extension of fuzzy set