The document discusses the origins and evolution of fuzzy logic, beginning with fuzzy set theory proposed by Zadeh in 1965 which aimed to represent vagueness in natural language using fuzzy sets with non-crisp boundaries. It explains key concepts in fuzzy logic like membership functions, fuzzy set operations, fuzzy relations and compositions. The document also compares classical sets with crisp boundaries to fuzzy sets and contrasts crisp logic with fuzzy logic which allows for degrees of truth between 0 and 1.

1. FUZZY LOGIC

2. Origins and Evolution of Fuzzy Logic
• Origin: Fuzzy Sets Theory (Zadeh, 1965)
• Aim: Represent vagueness and impre-cission
of statements in natural language
• Fuzzy sets: Generalization of classical sets
• In the 70s: From FST to Fuzzy Logic
• Nowadays: Applications to control systems
– Industrial applications
– Domotic applications, etc.

4. Classical Sets
 Classical sets – either an element belongs
to the set or it does not.
 For example, for the set of integers,
either an integer is even or it is not (it is
odd).

5. Classical Sets
Classical sets are also called crisp (sets).
Lists: A = {apples, oranges, cherries, mangoes}
A = {a1,a2,a3 }
A = {2, 4, 6, 8, …}
Formulas: A = {x | x is an even natural number}
A = {x | x = 2n, n is a natural number}
Membership or characteristic function
A
( x)
1 if x A
0 if x A

8. Fuzzy Sets
 Sets with fuzzy boundaries
A = Set of tall people
Crisp set A
Fuzzy set A
1.0
1.0
.9
Membership
.5
5’10’’
Heights
function
5’10’’
6’2’’
Heights

9. Membership Functions (MFs)
 Characteristics of MFs:
 Subjective measures
 Not probability functions

“tall” in Asia
MFs
.8

“tall” in the US
.5

“tall” in NBA
.1
5’10’’
Heights

10. Fuzzy Sets
 Formal definition:
A fuzzy set A in X is expressed as a set of ordered pairs:
A
Fuzzy set
{( x,
A
( x ))| x
Membership
function
(MF)
X}
Universe or
universe of discourse
A fuzzy set is totally characterized by a
membership function (MF).

11. An Example
• A class of students
(E.G. MCA. 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

12. Crisp or Fuzzy Logic
 Crisp Logic
 A proposition can be true or false only.
• Bob is a student (true)
• Smoking is healthy (false)
 The degree of truth is 0 or 1.
 Fuzzy Logic
 The degree of truth is between 0 and 1.
• William is young (0.3 truth)
• Ariel is smart (0.9 truth)

13.  Fuzzy Sets
 Membership
function
A
X
[0,1]
Crisp Sets
Characteristic
function
mA
X
{0,1}

15. Set-Theoretic Operations
 Subset
A
B
A
( x)
B
( x),
x
U
 Complement
A U
 Union
C
A
B
C
A
( x)
A
( x) 1
max(
A
( x),
min(
A
A
B
( x)
( x))
A
( x)
B
( x)
 Intersection
C
A
B
C
( x)
( x),
B
( x ))
A
( x)
B
( x)

16. Set-Theoretic Operations
A
B
A
A B
A B

17. Properties Of Crisp
Set
Involution
A
Commutativity
A B B
A B B
Associativity
Distributivity
Idempotence
Absorption
A
De Morgan’s laws
A
A
A
C
A
B C
A B
C
A
B C
A B C
A B C
A A A
A A A
A
A
A
A
B
B
A B
A B
A
A
A C
A C
A
B
A
A B
B
B
A
B

18. Properties
 The following properties are invalid for
fuzzy sets:
 The laws of contradiction
A
A
A
A U
 The laws of exclude middle

19. Properties of Fuzzy Set

20. 1
0
height
short
tall

21. Example
 we have two discrete fuzzy sets

22. Example (cont..)

23. Summarize properties
Involution
A
Commutativity
A
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
A (A B)=A, A (A B)=A
A A B A B
A A B A B
A X=X, A
=
Absorption of complement
Abs. by X and
Identity
A
=A, A X=A
Law of contradiction
A
A
Law of excl. middle
A
A X
DeMorgan’s laws
A
B
A
B
A
B
A
B

25. Fuzzy Set Operations

26. Crisp Set Operations

27. Representation of Crisp set

29. CARTESIAN PRODUCT
An ordered sequence of r elements, written in the
form (a1, a2, a3, . . . , ar), is called an ordered rtuple.
For crisp sets A1,A2, . . . ,Ar, the set of all rtuples(a1, a2, a3, . . . , ar), where a1∈A1,a2 ∈A2,
and ar∈Ar, is called the Cartesian product of
A1,A2, . . . ,Ar, and is denoted by
A1 A2 ··· Ar.
(The Cartesian product of two or more sets is not the
same thing as the arithmetic product of two or
more sets.)

30. Crisp Relations
A subset of the Cartesian product A1 A2 ··· Ar
is called an r-ary relation over A1,A2, . . . ,Ar.
If three, four, or five sets are involved in a subset
of the full Cartesian product, the relations are
called ternary, quaternary, and quinary

31. Cartesian product
The Cartesian product of two universes X and Y is
determined as
 X Y = {(x, y) | x ∈X, y ∈Y}
which forms an ordered pair of every x ∈X with
every y ∈Y, forming unconstrained
 matches between X and Y. That is, every element
in universe X is related completely to every
element in universe Y.

32. Fuzzy Relations
 Triples showing connection between two sets:
(a,b,#): a is related to b with degree #
 Fuzzy relations are set themselves
 Fuzzy relations can be expressed as matrices
…
32

33. Fuzzy Relations Matrices
 Example: Color-Ripeness relation for tomatoes
R1(x, y)
unripe
semi ripe
ripe
green
1
0.5
0
yellow
0.3
1
0.4
Red
0
0.2
1
33

34. Fuzzy Relations
A fuzzy relation R is a 2D MF:
R
( x, y),
R
( x, y) | ( x, y)
X Y

35. Fuzzy relation
 A fuzzy relation is a fuzzy set defined on the
Cartesian product of crisp sets A1, A2, ..., An
where tuples (x1, x2, ..., xn) may have varying
degrees of membership within the relation.
 The membership grade indicates the strength
of the relation present between the elements of
the tuple.
R
R
: A1 A2 ... An
(( x1 , x2 ,..., xn ),
[0,1]
R
)|
R
( x1 , x2 ,..., xn ) 0, x1
A1, x2
A2 ,..., xn
An
35

37. Max-Min Composition
X
Y
Z
R: fuzzy relation defined on X and Y.
S: fuzzy relation defined on Y and Z.
R。S: the composition of R and S.
A fuzzy relation defined on X an Z.
RS
(x, z) max y min
y
R
( x, y)
R
( x, y),
S
S
( y, z)
( y, z)

38.  Max-min composition
( x, y) A B, ( y, z) B C
max[min( R ( x, y ),
S R ( x, z )
y
y
[
R
( x, y )
S
S
( y, z ))]
( y, z )]
 Example
38

39.  Example
S R
(1,
) max[min(0.1, 0.9), min(0.2, 0.2), min(0.0, 0.8), min(1.0, 0.4)]
max[0.1, 0.2, 0.0, 0.4] 0.4
39

40.  Example
S R
(1,
)
max[min(0.1, 0.0), min(0.2,1.0), min(0.0, 0.0), min(1.0, 0.2)]
max[0.0, 0.2, 0.0, 0.2] 0.2
40

41. 41

42. Max-min composition is not mathematically tractable,
therefore other compositions such as max-product
composition have been suggested.
Max-Product Composition
X
Y
Z
R: fuzzy relation defined on X and Y.
S: fuzzy relation defined on Y and Z.
R。S: the composition of R and S.
A fuzzy relation defined on X an Z.
RS
(x, y) maxv
R
( x, v)
S
(v, y)

43. Max Product
 Max product: C = A・B=AB=
 Example
C12
?
43

44. Max product
 Example
C12
0.1
44

45. Max product
 Example
C13
0.5
45

46. Max product
 Example
C
46

48. Introduction
Fuzzify crisp inputs to get the fuzzy inputs
Defuzzify the fuzzy outputs to get crisp
outputs

49. Fuzzy Systems
Input
Fuzzifier
Inference
Engine
Fuzzy
Knowledge base
Defuzzifier
Output

50. Propositional logic
 A proposition is a statement- in which English
is a declarative sentence and logic defines
the way of putting symbols together to form
a sentences that represent facts
 Every proposition is either true or false.

51. Example of PL
The conjunction of the two sentences:
Grass is green
Pigs don't fly
is the sentence:
Grass is green and pigs don't fly
The conjunction of two sentences will be true if,
and only if, each of the two sentences from
which it was formed is true.

52. Statement symbols and
variables
 Statement:
A simple statement is one that does
not contain any other statement as a part.
A compound statement is one that has
two or more simple statement as parts called
components.

53. Symbols for connective
ASSERTION
P
NEGATION
“P IS TRUE”
¬P
~
!
CONJUCTION
P^Q
.
&
DISJUNCTION
PvQ
||
IMPLICATION
P-> Q
EQUIVALENCE P⇔Q
NOT
AND
“BOTH P AND Q ARE
TRUE
|
OR
“ EITHER P OR Q IS
TRUE”
⇒
=
&&
“P IS FALSE”
IF…THEN
“IF P IS TRUE THEN Q
IS TRUE.”
⇔
IF AND
ONLY IF
“P AND Q ARE EITHER
BOTH TRUE OR
FALSE”

54. Truth Value
 The truth value of a statement is truth or
falsity.
P is either true or false
~p is either true of false
p^q is either true or false, and so on.
 Truth table is a convenient way of showing
relationship between several propositions..

55. Truth Table for Negation
P
~P
Case 1
T
F
Case 2
F
T
As you can see “P” is a true statement then its
negation “~P” or “not P” is false.
If “P” is false, then “~P” is true.

56. Truth Table for Conjunction
P
Q
PΛQ
Case 1
T
T
T
Case 2
T
F
F
Case 3
F
T
F
Case 4
F
F
F

57. Truth Table for Disjunction
P
Q
PVQ
Case 1
T
T
T
Case 2
T
F
T
Case 3
F
T
T
Case 4
F
F
F

58. Tautology
 Tautology is a proposition formed by
combining other proposition (p,q,r…)which is
true regardless of truth or falsehood of p,q,r…
 DEF: A compound proposition is called a
tautology if no matter what truth values
its atomic propositions have, its own truth
value is T.

59. Tautology example
Demonstrate that
[¬p (p q )] q
is a tautology in two ways:
1. Using a truth table – show that
(p q )] q is always true.
[¬p
L3
59

60. Tautology by truth table
p q ¬p p q ¬p (p q ) [¬p (p q )] q
T T
T F
F T
F F
L3
60

61. Tautology by truth table
p q ¬p p q ¬p (p q ) [¬p (p q )] q
T T
F
T F
F
F T
T
F F
T
L3
61

62. Tautology by truth table
p q ¬p p q ¬p (p q ) [¬p (p q )] q
T T
F
T
T F
F
T
F T
T
T
F F
T
F
L3
62

63. Tautology by truth table
p q ¬p p q ¬p (p q ) [¬p (p q )] q
T T
F
T
F
T F
F
T
F
F T
T
T
T
F F
T
F
F
L3
63

64. Tautology by truth table
p q ¬p p q ¬p (p q ) [¬p (p q )] q
T T
F
T
F
T
T F
F
T
F
T
F T
T
T
T
T
F F
T
F
F
T
L3
64

65. Modus Ponens and Modus
Tollens
 Modus ponens -- If A then B, observe A,
conclude B
 Modus tollens – If A then B, observe notB, conclude not-A

66. Modus Ponens and Tollens
 If Joan understood this book, then she would
get a good grade. If P then Q
 Joan understood .: she got a good grade.
 This uses modus ponens.
P .: Q
 If Joan understood this book, then she would
get a good grade. If P then Q
 She did not get a good grade .: she did not
understand this book.
~Q .: ~P
 This uses modus tollens.

67. Fuzzy Quantifiers
The scope of fuzzy propositions can be
extended using fuzzy quantifiers
• Fuzzy quantifiers are fuzzy numbers that take
part in fuzzy propositions
• There are two different types:
– Type #1 (absolute): Defined on the set of real
numbers
• Examples: “about 10”, “much more than 100”, “at
least
about 5”, etc.
– Type #2 (relative): Defined on the interval [0, 1]
• “almost all”, “about half”, “most”, etc.

68. Fuzzification
 The fuzzification comprises the process of
transforming crisp values into grades of
membership for linguistic terms of fuzzy sets.
The membership function is used to associate
a grade to each linguistic term.
 Measurement devices in technical systems
provide crisp measurements, like 110.5 Volt or
31,5 C. At first, these crisp values must be
transformed into linguistic terms (fuzzy sets) .
This is called fuzzification.

69. Input
Fuzzifier
Fuzzifier
Inference
Engine
Defuzzifier
Output
Fuzzy
Knowledge base
Converts the crisp input to a linguistic variable using
the membership functions stored in the fuzzy
knowledge base.

70. Fuzzy interference
If x is A and y is B then z = f(x, y)
Fuzzy Sets
Crisp Function
f(x, y) is very often a polynomial
function w.r.t. x and y.

71. Examples
R1: if X is small and Y is small then z = x +y +1
R2: if X is small and Y is large then z = y +3
R3: if X is large and Y is small then z = x +3
R4: if X is large and Y is large then z = x + y + 2

73. Defuzzification
 • Convert fuzzy grade to Crisp output
 The max criterion method finds the point at
which the membership function is a
maximum.
 The mean of maximum takes the mean of those
points where the membership function is at a
maximum.

74. Defuzzification