Fuzzy Set Theory
Basic concept:
Membership function:
U, X is a subset of U, we define a membership function for X:
e.g.: U = {a, b, c, d, e}
X1 = {a/0.3, b/0.2, c/0.5}
X2 = {a/0.2, b/0.1, d/0.5, e/0.2}
X1 intersect X2 = {a/0.2, b/0.1}
X1 union X2 = {a/0.3, b/0.2, c/0.5, d/0.5, e/0.2}
~X1(e) = 1 – muX1(e) = {a/0.7, b/0.8, c/0.5, d/1, e/1}
X1 union ~X1 = {a/0.7, b/0.8, c/0.5, d/1, e/1}
Fuzzy Relations:
Binary fuzzy relations:
Basic concept: U1 and U2 are universes,
A binary relation R is a subset of U1 X U2
A fuzzy binary relation is determined by the membership function  :
 : U1 X U2  [0, 1]
Operations on fuzzy binary relations:
1. Projection: max operator
2. Extension: Cylindrical extension
3. Composition: max-min composition
e.g.:
X1 = {a/0.2, b/0,3, c/0.5}
X2 = {a/0.3, b/0.3, c/0.4}
R1
a
a
0.3
b
0.3
c
0.5
b
0.3
0.3
0.5
c
0.4
0.4
0.5
Cylindrical extension R11
a
a
0.5
b
0.5
c
0.5
b
0.5
0.5
0.5
c
0.5
0.5
0.5
Cylindrical extension R12
a
a
0.4
b
0.4
c
0.5
b
0.4
0.4
0.5
c
0.4
0.4
0.5
b
0.4
0.4
0.5
c
0.4
0.4
0.5
b
0.5
0.5
0.5
c
0.5
0.5
0.5
R11  R12
a
b
c
a
0.4
0.4
0.5
R1  (R11  R12)
Cylindrical extension R11
a
a
0.5
b
0.5
c
0.5
Cylindrical extension R12
a
a
0.4
b
0.4
c
0.5
b
0.4
0.4
0.5
c
0.4
0.4
0.5
R11  R12 (Max-min Composition)
a
a
0.5
b
0.5
c
0.5
b
0.5
0.5
0.5
c
0.5
0.5
0.5
Fuzzy Rule-Based Systems:
A: scores of projects [0, 100]
B: final Scores
A >= 90: [90, 100]
B >= 90: [90, 100]
If A >= 90 then B >= 90
Steps:
1. Identify variables V1, V2, …, Vm in an application
2. Determine domains Di’s of those variables (Universe of
discourse)
3. Define fuzzy sets on Di’s: for each Di we associate a family of
fuzzy sets, called reference fuzzy sets.
e.g.: for a temperature variable T, we can define the fuzzy sets:
Hot, Warm, Cool, Cold
e.g.: Temp = [-100, 100]
Fuzzification process: is a mapping of variable domains into fuzzy
sets.
Linguistic Variables: variables whose values are fuzzy sets.
e.g.: the domain of T is the set {Hot, Warm, Cool, Cold}.
Now, the value of T may be Hot or Cold, etc.
Domain(Speed) = {Stop, Slow, Medium, Fast}.
4. Write fuzzy rules using the linguistic variables obtained from
Step 3.
e.g.:
if T is Cold then Speed is Stop
if T is Cool then Speed is Slow
if T is Warm then Speed is Medium
if T is Hot then Speed is Fast
5. Rule of inference:
6. Defuzzification Methods: Centroid or average
e.g:
X = {2,3,4,5,6,7,8,9} Y = {1,2,3,4,5,6}
Medium = {0.1/2 + 0.3/3 + 0.7/4 + 1/5 + 1/6 + 0.7/7 + 0.5/8 + 0.2/9}
Small = {1/1 + 1/2 + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6}
If x is Medium then y is Small
Use min Cartesian product, the rule is represented by the following
relation R:
0.1
0.1
0.1
0.1
0.1
0.1
0.3
0.3
0.3
0.3
0.3
0.1
0.7
0.7
0.7
0.6
0.3
0.1
1
1
0.9
0.6
0.3
0.1
1
1
0.9
0.6
0.3
0.1
0.7
0.7
0.7
0.6
0.3
0.1
0.5
0.5
0.5
0.5
0.3
0.1
0.2
0.2
0.2
0.2
0.2
0.1
X is Small = {1/2 + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6}
[1 0.9 0.6 0.3 0.1 0 0 0] o R = [0.6 0.6 0.6 0.6 0.3 0.1]
(1*0.6 + 2 *0.6 + 3 * 0.6 + 4 * 0.6 + 5 * 0.3 + 6 * 0.1) / (0.6 + 0.6 + 0.6
+ 0.6 + 0.3 + 0.1)
= 8.1/2.8 = 2.89