# fuzzy set

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Fuzzy Rendszerek I.
Dr. K&oacute;czy L&aacute;szl&oacute;
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An Example
• A class of students
(E.G. M.Sc. Students taking „Fuzzy Theory”)
• The universe of discourse: X
• “Who does have a driver’s licence?”
• A subset of X = A (Crisp) Set
• (X) = CHARACTERISTIC FUNCTION
1
0
1
1
0
1
1
• “Who can drive very well?”
(X) = MEMBERSHIP FUNCTION
0.7
0
1.0
0.8
0
0.4
0.2
FUZZY SET
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History of fuzzy theory
•
•
•
•
Fuzzy sets &amp; logic: Zadeh 1964/1965Fuzzy algorithm: Zadeh 1968-(1973)Fuzzy control by linguistic rules: Mamdani &amp; Al. ~1975Industrial applications: Japan 1987- (Fuzzy boom), Korea
Home electronics
Vehicle control
Process control
Pattern recognition &amp; image processing
Expert systems
Military systems (USA ~1990-)
Space research
• Applications to very complex control problems: Japan 1991E.G. helicopter autopilot
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An application example
One of the most interesting applications of fuzzy computing:
“FOREX” system.
1989-1992, Laboratory for International Fuzzy Engineering Research
(Yokohama, Japan) (Engineering – Financial Engineering)
To predict the change of exchange rates (FOReign EXchange)
~5600 rules like:
“IF the USA achieved military successes on the past day [E.G. in the
Gulf War] THEN &yen;/\$ will slightly rise.”
Inputs
(Observations)
FOREX
Prediction
Fuzzy Inference Engine
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Another Example
&yen;/\$
100
1993
1994
1995
Time
What is fuzzy here?
- What is the tendency of the &yen;/\$ exchange rate?
“It’s MORE OR LESS falling” (The general tendency is “falling”, there’s
no big interval of rising, etc.)
- What is the current rate?
Approximately 88 &yen;/\$  Fuzzy number
- When did it first cross the magic 100 &yen;/\$ rate? SOMEWHEN in mid
1995
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A complex problem
Many components, very complex
system. Can AI system solve it?
Not, as far as we know. But WE
can.
Our car, save fuel, save time, etc.
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Definitions
• Crisp set:
c•
•d
•a
•x
•y
aA
bA
B
• Convex set:
Crisp set A
A is not convex as aA, cA, but
d=a+(1-)c A, [0, 1].
B is convex as for every x, yB and
[0, 1] z=x+(1-)y B.
• Subset:
•x
•y
A
B
If xA then
also xB.
AB
•b
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Definitions
• Equal sets:
If AB and BA then A=B if not so AB.
• Proper subset:
If there is at least one yB such that yA then AB.
• Empty set: No such x.
• Characteristic function:
A(x): X{0, 1}, where X the universe.
0 value: x is not a member,
1 value: x is a member.
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Definitions
A={1, 2, 3, 4, 5, 6}
• Cardinality: |A|=6.
• Power set of A:
P (A)={{}=&Oslash;, {1}, {2}, {3}, {4}, {5}, {6}, {1, 2}, {1, 3}, {1, 4},
{1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6},
{4, 5}, {4, 6}, {5, 6}, {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 2, 6}, {1, 3,
4}, {1, 3, 5}, {1, 3, 6}, {1, 4, 5}, {1, 4, 6}, {1, 5, 6}, {2, 3, 4}, {2,
3, 5}, {2, 3, 6}, {2, 4, 5}, {2, 4, 6}, {2, 5, 6}, {3, 4, 5}, {3, 4, 6},
{3, 5, 6}, {4, 5, 6}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 3, 6}, {1, 2, 4,
5}, {1, 2, 4, 6}, {1, 2, 5, 6}, {1, 3, 4, 5}, {1, 3, 4, 6}, {1, 3, 5, 6},
{1, 4, 5, 6}, {2, 3, 4, 5}, {2, 3, 4, 6}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 4,
5, 6}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6}, {1, 2, 4, 5, 6}, {1, 3, 4, 5, 6},
{2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6}}.
|P (A)|=2|6|=64.
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Definitions
• Relative complement or difference:
A–B={x | xA and xB}
B={1, 3, 4, 5}, A–B={2, 6}.
C={1, 3, 4, 5, 7, 8}, A–C={2, 6}!
• Complement: A  X  Awhere X is the universe.
Complementation is involutive: A  A
Basic properties:   X,
• Union:
X
AB={x | xA or xB}
n
For
Ai | i  I  Ai  x | x  Ai for some i
AX  X
A   A
AA  X
i 1
(Law of excluded middle)
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Definitions
• Intersection:
AB={x | xA and xB}.
n
For
Ai | i  I  Ai  x | x  Ai for all i
A   
AX  A
i 1
AA  
• More properties:
Commutativity:
Associativity:
Idempotence:
Distributivity:
AB=BA, AB=BA.
ABC=(AB)C=A(BC),
ABC=(AB)C=A(BC).
AA=A, AA=A.
A(BC)=(AB)(AC),
A(BC)=(AB)(AC).
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Definitions
• More properties (continued):
DeMorgan’s laws:
AB  AB
AB  AB
• Disjoint sets: AB=.
• Partition of X:


x   A i | i  I, A i1  A i 2  ,  A i  X 


iI
6
X   Ai
i 1
Ai  A j  
A i | i  N 6   x 
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Summarize properties
AA
Involution
Commutativity
AB=BA, AB=BA
Associativity
ABC=(AB)C=A(BC),
ABC=(AB)C=A(BC)
Distributivity
A(BC)=(AB)(AC),
A(BC)=(AB)(AC)
Idempotence
AA=A, AA=A
Absorption
Absorption of complement
Abs. by X and 
Identity
A(AB)=A, A(AB)=A
A  A  B  A  B
A  A  B  A  B
AX=X, A=
A=A, AX=A
AA  
Law of excl. middle
AA  X
DeMorgan’s laws
A B  A B
A B  A B
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Membership function
A 
Crisp set
Fuzzy set
Characteristic function
Membership function
A:X{0, 1}
A:X[0, 1]
1
1  5 x 102


0
x  5  x  17



 B   0.2x  5
5  x  10 
 1

 7 x  17  10  x  17 
 C  2A
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Membership function (2)
• More general Idea:
L-Fuzzy set, A: XL, L has a linear (full) or partial ordering.
• Vector valued fuzzy set:
A: x[0, 1]n=[0,1]…[0,1] (n times).
• Fuzzy measure:
g: P (x)[0, 1]
Degree of belonging to a crisp subset of the universe.
Example:
’teenager’=13..19 years old, ‘twen’=20..29 years old,
‘in his thirties’=30..39 years old, etc.
X={Years of men’s possible age}
‘teenager’X
g(‘Susan is a twen’)=0.8,
g(‘Susan is a teenager’)=0.2,
g(‘Susan is in her thirties’)=0.1.
Best guess has highest measure.
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Definitions
~
 : X  P 0,1
• Fuzzy set of type 2:

A
2

1
1
P~
0
X1
• Level N:
X2
X
 : P~k -1 ( x )  0,1
~
~
~
k
P ( x )  P (P k -1 ( x ) ) k  1
~
P 0 (x)  x
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Definitions
• Interval valued fuzzy set:
 : X  P~ 0,1
even  : X  C 0,1
• C(Y) is the family of convex subsets of Y, i.e. 
P(Y)
• C([0,1]) is the family of all intervals
1

  (x)
0
• e.g.:

  ( x )  [0.9,1.0]
X
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Definitions
• Relation of various classes of sets discussed
L-Fuzzy
Type N
Type N-1
Type 2
Interval V’D
Fuzzy Set
Vector V’D
Crisp Set
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Some basic concepts of fuzzy sets
Elements
Young
Old
5
0
0
1
0
10
0
0
1
0
20
0
.8
.8
.1
30
0
1
.5
.2
40
0
1
.2
.4
50
0
1
.1
.6
60
0
1
0
.8
70
0
1
0
1
80
0
1
0
1
Some basic concepts of fuzzy sets
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• Support: supp(A)={x | A(x)&gt;0}.
~
P
supp: x   P x 
Infant=0, so supp(Infant)=0.
If |supp(A)|&lt;, A can be defined
A=1/x1+ 2/x2+…+ n/xn.
n
A   i / x i
i 1
• Kernel (Nucleus, Core):
Kernel(A)={x | A(x)=1}.
A    A x  / x
x
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Definitions
• Height:
Height (A)  max ( A ( x ))  sup ( A ( x ))
x
• height(old)=1
– If height(A)=1
– If height(A)&lt;1
– height(0)=0
height(infant)=0
A is normal
A is subnormal
• a-cut: A a  {x |  A ( x )  a}
x
Strong Cut: A a  {x |  A ( x )  a}
Young 0.8  {5,10}
• Young 0.8  {5,10, 20}
– Kernel:
A1  {x |  A ( x )  1}
– Support: A 0  {x |  A ( x )  0}
• If A is subnormal, Kernel(A)=0
– A a  A IF
a
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Definitions
• Level set of A:
 A  {a |  A ( x )  a for some x  X}
• Convex fuzzy set:
X  n
– A is convex if for every x,yX and
 [0,1]:
 A (x  (1  ) y)  min(  A ( x ),  A ( y))
– All sets on the previous figure are
convex FS’s
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Definitions
• Nonconvex fuzzy set:
• Convex fuzzy set over R 2
x2
Kernel
a=0.2
a=0.4
a=0.8
a=0.6
x1
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Definitions
• Fuzzy cardinality of FS A:
~
Fuzzy number A
~  A a   a for all aA
A
~
O l d  0.1 / 7  0.2 / 6  0.4 / 5  0.6 / 4  0.3 / 3  1 / 2
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Definitions
• Fuzzy number: Convex and normal fuzzy set of 
– Example 1:
N
1
0
r2

height(N(r))=1
for any r1, r2: N(r*)= N(r1+(1- )r2)  min(N(r1), N(r2))
– Example 2: “Approximately equal to 6”
r1
r*

r6
1 
if r  [3, 9]
 N (r )  
3
1
0 otherwise

N(r)
0
3
6
9

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Definitions
• Flat fuzzy number:
There is a,b (ab, a,b) N(r)=1 IFF r[a,b]
(Extension of ‘interval’)
• Containment (inclusion) of fuzzy set
AB IFF A(x) B(x)
– Example:
• Equal fuzzy sets
A=B IFF AB and AB If it is not the case: AB
• Proper subset
AB IFF AB and A  B
– Example:
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Definitions
• Extension principle:
How to generalize crisp concepts to fuzzy?
Suppose that X and Y are crisp sets
X={xi}, Y={yi} and f: XY
~
If given a fuzzy set A  P x 
n
~
A   i / xi THEN B  P (Y ) IS
i 1
 n
 n
B  f  A  f   i / xi    i / f  xi 
 i 1
 i 1
 


The latter is understood that if for
yi  f xij FOR xij | j n THEN
  yi   max  x j | f x j   yi 
IF
   THEN   y   0
i
xj
i
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Definitions
a  d

f  b  f
c  d

– Example
X={a,b,c}
Y={d,e,f}
 A  X   .1/ a  .5 / b  .8 / c
 f  A Y   .8 / d  .5 / f
• Arithmetics with fuzzy numbers:
Using extension principle
a+b=c
(a,b,c  )
A+B=C


~
~
 A, B  P  EVEN C  


WHAT IS C ?



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
1
0
Example
0 1
2
A
B
3
6
4 5
C n   max
a ,b
n
a
b
min(A,B)
3
0
3
0
4
0
1
4
3
0
0
5
0
1
2
5
4
3
0
0.5
0
6
0
1
2
3
6
5
4
3
0
0.5
0.5
0
1
...
7
6
0
0.5
7
C

7 8 9 10 11 12
min  A a ,  B b  | a  b  n 
n
a
b
min(A,B)
7
2
3
4
5
4
3
1
0.5
0
8
1
2
3
4
7
6
5
4
0
0.5
0.5
0
9
2
3
4
7
6
5
0
0.5
0
10
3
4
7
6
0
0
11
4
7
0
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Definitions
• Fuzzy set operations defined by L.A. Zadeh in
1964/1965
• Complement:  A x   1   A x 
AB x   min A x , B x 
• Intersection:
 AB x   max  A x , B x 
• Union:
(x):
 x  A, B
0
 x  A, B
1
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Definitions
This is really a generalization of crisp set op’s!
A
B
A
0
0
1
1
0
1
0
1
1
1
0
0
AB AB
0
0
0
1
0
1
1
1
1-A
min
max
1
1
0
0
0
0
0
1
0
1
1
1
Classical (Two valued) logic
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• Classical (two valued logic):
Any logic – Means of a reasoning propositions: true
(1) or false (0). Propositional logic uses logic
variables, every variable stands for a proposition.
When combining LV’s new variables (new
propositions) can be constricted.
Logical function:
f: v1, v2, …, vn  vn+1
2n
• 2 different logic functions of n variables exist.
E.G. if n=2, 16 different logic functions (see next
page).
Logic functions of two variables
v2
v1
1100
1010
Symbol
Other names used
w1
0000
Zero fn.
0
Falsum
w2
0001
Nor fn.
v1v2
Pierce fn.
w3
0010
Inhibition
v1&gt;v2
Proper inequality
w4
0011
Negation
v2
Complement
w5
0100
Inhibition
v1&lt;v2
Proper inequality
w6
0101
Negation
v1
Complement
w7
0110
Excl. or fn.
v1v2
Antivalence
w8
0111
Nand fn.
v1v2
Sheffer Stroke
w9
1000
Conjunction
v1v2
And function
w10
1001
Biconditional
v1v2
Equivalence
w11
1010
Assertion
v1
Identity
w12
1011
Implication
v1v2
Conditional, ineq.
w13
1100
Assertion
v2
Identity
w14
1101
Implication
v1v2
Conditional, ineq.
w15
1110
Disjunction
v1v2
Or function
w16
1111
One fn.
1
Verum
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1 / 34
Definitions
• Important concern:
How to express all logic fn’s by a few logic
primitives (fn’s of one or two lv’s)?
A system of lp’s is (functionally) complete if all
LF’s can be expressed by the FNs in the system.
A system is a base system if it is functionally
complete and when omitting any of its elements
the new system isn’t functionally complete.
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Definitions
• A system of LFN’s is functionally complete if:
–
–
–
–
–
At least one doesn’t preserve 0
At least one doesn’t preserve 1
At least one isn’t monotonic
At least one isn’t self dual
At least one isn’t linear
• Example:
AND , NOT
A
00
A
1 1
A
0  1, 1  0, 0  1, 1  0
A  B A  B  A  B  A  B 
A  B CANNOT BE EXPRESSED
BY ONLY 
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Definitions
• Importance of base systems, and functionally
complete systems.
Digital engineering:
Which set of primitive digital circuits is suitable to
construct an arbitrary circuit?
• Very usual: AND, OR, NOT (Not easy from the point
of view of technology!)
• NAND (Sheffer stroke)
• NOR (Pierce function)
• NOT, IMPLICATION (Very popular among logicians.)
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Definitions
• Logic formulas:
– E.G.: Let’s adopt +,-,* as a complete system. Then:
• 0 and 1 are LFs
• If v is a LF v is a LF
• If A and B are LFs, then A+B, A*B are LFs
• There are no other LFs
– Similarly with -,, etc.
• There are infinitely many ways to describe a LF in an
equivalent way
– E.G.: A, A, A+A, A*A, A+A+A, etc.
• Canonical formulas, normal form
– DCF AB  AB  AB  A  B 
– CCF A  B A  B   AB  AB 
– Logic formulas with identical truth values are named
equivalent
a=b
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Definitions
• Always true logic formulas:
Tautology:
AA+A
• Always false logic formulas:
AA
• Various forms of tautologies are used for didactive
inference = inference rules
– Modus Ponens:
(a*(ab))b
– Modus Tollens:
b * a  b  a
– Hypothetical Syllogism:
((a b)*(b c)) (a c)
– Rule of Substitution:
If in a tautology any variable is replaced by a LF then it
remains a tautology.
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Definitions
• These inference rules form the base of expert
systems and related systems (even fuzzy control!)
• Abstract algebraic model:
Boolean Algebra
B=(B,+,*,-)
B has at least two different elements (bounds): 0
and 1
+ some properties of binary OPs “+” and “*”, and
unary OP “-”.
Properties of boolean algebras
B1. Idempotence
a+a=a, a&middot;a=a
B2. Commutativity
a+b=b+a
B3. Associativity
(a+b)+c=a+(b+c),
(a&middot;b)&middot;c=a&middot;(b&middot;c)
B4. Absorption
a+(a&middot;b)=a, a&middot;(a+b)=a
B5. Distributivity
a&middot;(b+c)=(a&middot;b)+(a&middot;c),
a+(b&middot;c)=(a+b)&middot;(a+c)
B6. Universal bounds
a+0=a, a+1=1
a&middot;1=a, a&middot;0=0
B7. Complementarity
a+a=1, a&middot;a=0, 1=0
B8. Involution
(a+b)= a &middot; b
B9.Dualization
(a&middot;b)= a +b
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Lattice
1 / 41
Definitions
Set
theory
P(X)
Boolean
algebra
B
Propositional
logic
F(V)

+


*

-
-
-
X
1
1

0
0



• Correspondences
defining
isomorphisms
between set theory,
boolean algebra and
propositional logic
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Definitions
• Isomorphic structure of crisp set and logic
operations = boolean algebra
Lattice
B.A.
• Structure of propositions:
x is P
–
–
Dr. K&oacute;czy is above

 190cm




x1
SUBJECT
x2=Dr.Kim
P
PREDICATE TRUE
P(x2)=FALSE!
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Definitions
• Quantifiers:
– (x) P(x):
There exists an x such
that x is P
– (x) P(x): For all x, x is P
– (!x) P(x): x and only one x such that x is P
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Definitions
• Two valued logic questioned since B.C.
Three valued logic includes indeterminate value: &frac12;
Negation: 1-a,  differ in these logics.
• Examples:
ab
Łukasiewicz
Bochvar
Kleene
Heyting
Reichenbach





00
0011
0011
0011
0011
0011
0&frac12;
0&frac12;1&frac12;
&frac12;&frac12;&frac12;&frac12;
0&frac12;1&frac12;
0&frac12;10
0&frac12;1&frac12;
01
0110
0110
0110
0110
0110
&frac12;0
0&frac12;&frac12;&frac12;
&frac12;&frac12;&frac12;&frac12;
0&frac12;&frac12;&frac12;
0&frac12;00
0&frac12;&frac12;&frac12;
&frac12;&frac12;
&frac12;&frac12;11
&frac12;&frac12;&frac12;&frac12;
&frac12;&frac12;&frac12;&frac12;
&frac12;&frac12;11
&frac12;&frac12;11
&frac12;1
&frac12;11&frac12;
&frac12;&frac12;&frac12;&frac12;
&frac12;11&frac12;
&frac12;11&frac12;
&frac12;11&frac12;
10
0100
0100
0100
0100
0100
1&frac12;
&frac12;1&frac12;&frac12;
&frac12;&frac12;&frac12;&frac12;
&frac12;1&frac12;&frac12;
&frac12;1&frac12;&frac12;
&frac12;1&frac12;&frac12;
11
1111
1111
1111
1111
1111
1 / 45
Definitions
• No difference from classical logic for 0 and 1.
But:
a  a  0, a  a  1
are not true!
• Quasi tautology: doesn’t assume 0. Quasi
• Next step?
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N-valued logic
• N-valued logic:
1
n2 

Tn  0,
, ... ,
,1
n 1 
 n 1
Degrees of truth
(Lukasiewicz, ~1933)
a 1- a
a  b  min a, b 
a  b  max a, b 
a  b  min(1,1  b - a)
a  b 1- a - b
LOGIC
PRIMITIVES
1 / 47
Definitions
• Ln n=2, … , 

(0)
Classical Logic
 Rational truth
values
If T is extended to [0,1] we obtain (T ) L1 with continuum
1
truth degrees
• L1 is isomorphic with fuzzy set
It is enough to study one of them, it will reveal all the facts
above the other.
• Fuzzy logic must be the foundator of approximate
reasoning, based on natural language!
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Fuzzy Proportion
• Fuzzy proportion: X is P
‘Tina is young’, where:
‘Tina’: Crisp age, ‘young’: fuzzy predicate.
Fuzzy sets expressing
linguistic terms for ages
Truth claims – Fuzzy sets
over [0, 1]
• Fuzzy logic based approximate reasoning
is most important for applications!
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