1. Intro. ANN & Fuzzy Systems
Lecture 31
Fuzzy Set Theory (3)
2. Intro. ANN & Fuzzy Systems
(C) 2001-2003 by Yu Hen Hu 2
Outline
• Fuzzy Relation Composition and an Example
• Fuzzy Reasoning
3. Intro. ANN & Fuzzy Systems
(C) 2001-2003 by Yu Hen Hu 3
Fuzzy Relation Composition
• Let R be a fuzzy relation in X Y, and S be a fuzzy
relation in Y Z.
• The Max-Min composition of R and S, RoS, is a fuzzy
relation in X Z such that
RoS µRoS(x,z) = {µR(x,y) µS(y,z) }
= Max. {Min. {µR(x,y), µS(y,z)}}/(x,z)
• The Max-Product Composition of R and S, RoS, is a
fuzzy relation in X Z such that
RoS µRoS(x,z) = {µR(x,y) µS(y,z) }
= Max. {µR(x,y) µS(y,z)}/(x,z)
4. Intro. ANN & Fuzzy Systems
(C) 2001-2003 by Yu Hen Hu 4
Fuzzy Composition Example
• Let the two relations R and S be, respectively:
• The goal is to compute RoS using both Max-min and
Max-product composition rules.
R y1
y2
y3
S z1
z2
x1
0.4 0.6 0 y1
0.5 0.8
x2
0.9 1 0.1 y2
0.1 1
y1
0 0.6
7. Intro. ANN & Fuzzy Systems
(C) 2001-2003 by Yu Hen Hu 7
Fuzzy Reasoning
• Comparing crisp logic inference and fuzzy logic inference
Translation –
Age(Mary) = 22
(Age(Dana),Age(Mary)) = Age(Dana)–Age(Mary) = 3
Age(Dana) = Age(Mary) + 3 = 22 + 3 = 25
Crisp
logic
Mary is 22 years old
Dana is 3 years older than Mary .
Dana is (22 + 3) years old
8. Intro. ANN & Fuzzy Systems
(C) 2001-2003 by Yu Hen Hu 8
Fuzzy Reasoning
Fuzzy
logic
Mary is Young
Dana is much older than Mary .
Dana is (Young o Much_older)
Translation –
Age(Mary) = Young (Young is a fuzzy set)
(Age(Dana),Age(Mary)) = Much_older (a relation)
Age(Dana) = Young o Much_older
– a composite relation!
9. Intro. ANN & Fuzzy Systems
(C) 2001-2003 by Yu Hen Hu 9
Fuzzy Reasoning (cont'd)
• µAge(Dana)(x) = {µyoung(y) µmuch_older(x,y) }
The universe of discourse (support) is "Age" which may
be quantified into several overlapping fuzzy (sub)sets:
Young, Mid-age, Old with the following definitions:
Age
Young Mid-age Old
20 35 50
µ(Age)
5
10. Intro. ANN & Fuzzy Systems
(C) 2001-2003 by Yu Hen Hu 10
Fuzzy Reasoning (cont'd)
• Much_older is a relation which is defined as:
µmuch_older(x,y) = ,
.
0
20
0
)
(
20
1
,
20
1
y
x
y
x
y
x
y
x
0
10
20
30
40 y
x
10
20
30
40
µ (x,y)
much_older
11. Intro. ANN & Fuzzy Systems
(C) 2001-2003 by Yu Hen Hu 11
Reasoning Example
For each fixed x, find
µAge(Dana)(x) = max(min(µyoung(y),µmuch_older(x,y)):
0
0.2
0.4
0.6
0.8
0 5 10 15 20 25 30 35 40 45
1.0
x
µ (x)
Age(Dana)