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1. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
FUZZY LOGIC SYSTEMS

2. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
Point a is clearly a member of crisp set A; point b is unambiguously not
a member of set A.
Figure b shows the vague, ambiguous boundary of a fuzzy set A on same Universe
In the central (unshaded) region of the fuzzy set, point a is clearly a full member of
the set.
Outside the boundary region of the fuzzy set, point b is clearly not a member of the
fuzzy set.
However, the membership of point c, which is on the boundary region, is ambiguous
CLASSICAL SETS AND FUZZY SETS

3. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
 If complete membership in a set (such as point a ) is
represented by the number 1, and no-membership in a set
(such as point b ) is represented by 0 , then point c must
have some intermediate value of membership (partial
membership in fuzzy set A ) on the interval [0,1].
 Presumably, the membership of point c in A approaches
a value of 1 as it moves closer to the central (unshaded)
region
and
 the membership of point c in A approaches a value of 0 as
it tries to leave the boundary region of A
CLASSICAL SETS AND FUZZY SETS

4. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
 Crisp set X, is a collection of objects all having the
same characteristics.
 The individual elements in the universe X will be
denoted as x
 Examples of elements of various universes might be as
follows:
 the clock speeds of computer CPUs;
 the operating currents of an electronic motor;
 the operating temperature of a heat pump (in degrees Celsius);
 the Richter magnitudes of an earthquake;
 the integers 1 to 10.
CLASSICAL SETS AND FUZZY SETS

5. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
Example: FUZZY SETS

6. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
Examples of fuzzy sets
 High temperature
 Low Pressure
 Colour of Apple
 Sweetness of Orange
 Weight of mango
 Degree of memebership value lie n range [0….1]

7. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
CONCEPT OF FUZZY SYSTEM
Input and output are crisp values : Fuzzification and defuzzification

8. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
CRISP SETS: AN OVERVIEW
General conventions used
 Z = {..., -2, -1, 0, 1, 2, ...} (the set of all integers),
 N = {1, 2, 3,…} (the set of all positive integers or natural
numbers),
 N+ = {0, 1, 2, ...} (the set of all nonnegative integers),
 N- = {0,-1,-2 ,-3,n -n},
 R: the set of all real numbers,
 R+: the set of all nonnegative real numbers,

9. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
 [a, b]: closed interval,
 (a, b]: left-open interval,
 [a, b): right-open interval
 (a, b): open interval of real numbers between a and b,
respectively,
 〈x1, x2, ... , xn〉 : ordered n-tuple of elements x1, x2, ... ,
xn.
 "iff" is a shorthand expression of "if and only if,"
 ∈ : existential quantifier
 ⩝ : the universal quantifier
CRISP SETS: AN OVERVIEW

10. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
 X or U denotes the universe of discourse, or universal
set.
 ∅: Empty set or Null set
 If x is a member or element of a set A,
 x is not an element of a set A
CRISP SETS: AN OVERVIEW

11. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
 1. List method: naming all the members of the set .
used only for finite sets.
Set A, whose members are a1, a2, ... , a,,, is usually
written as
 2. Rule Method: Define the set by a property satisfied
by all its members
all elements of X for which the proposition P(x) is true.
P(x) designates a proposition of the form "x has the property P.
the symbol | denotes the phrase "such that,"
CRISP SETS: AN OVERVIEW

12. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
3. Characteristic function : Define set by a function that declares
which elements of X are members of the set and which are
not.
Example : Set A is defined by its characteristic function,
as follows:
The characteristic function maps
elements of X to elements of the set {0, 1}
CRISP SETS: AN OVERVIEW
the transition for an element in the universe
between membership and non-membership in a
given set is abrupt and well defined

13. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
CRISP SETS: AN OVERVIEW

14. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
CRISP SETS: AN OVERVIEW

15. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
power set , P(X) : constitutes all possible sets of X
Let X={a,b,c} cardinality |X|= 3
Norte that if the cardinality of the universe is infinite,
then the cardinality of the power set is also infinity,
that is,
|X |=∞ |P(X)| =∞.
CRISP SETS: AN OVERVIEW

16. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
 UNION:

 INTERSECTION:
 COMPLEMENT:
OPERATIONS ON CRISP SETS

17. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
DIFFERENCE
OPERATIONS ON CRISP SETS

18. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
PROPERTIES OF CRISP SETS

19. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
PROPERTIES OF CRISP SETS

20. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
 Crisp set theory is not capable of representing
descriptions and classifications in many cases;
 Crisp Sets do not provide adequate representation
Limitation of crisp sets

21. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
MOTIVATION FOR Non-CRISP OR
FUZZY SETS
Linguistic variables are often used to describe, and maybe
classify, physical objects and situations.

22. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
 In real world there exists much of fuzzy knowledge
 Fuzzy knowledge : knowledge that is vague,
imprecise, uncertain, ambiguous, inexact or
probabilistic in nature
MOTIVATION FOR Non-CRISP OR
FUZZY SETS

23. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
MOTIVATION FOR Non-CRISP OR
FUZZY SETS
The proposition of Fuzzy Sets are motivated by the need
to capture and represent real world data with
uncertainty due to imprecise measurement.
 − The uncertainties are also caused by vagueness in the
language.(linguistic variables)

24. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
Non CRISP OR FUZZY SETS
 The characteristic function of the crisp set is
generalized for the Non-crisp sets. This
generalized characteristic function is called
membership function.
the transition for an element in the universe between membership and non-
membership in a given set is gradual conforming to the fact that the
boundaries of the fuzzy sets are vague and ambiguous. Hence, membership
of an element from the universe in this set is measured by a function that
attempts to describe vagueness and ambiguity.
Example : How is weather today?
Is XYZ person honest?

25. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
 . For a given crisp set A the characteristics function
assigns a value μA(x) to every x Є X such that
μA(x) = 0 if x ЄA
=1 if x
 the function maps elements of the universal set to
the set containing 0 and 1 µA : X < {0, 1}
Non CRISP OR FUZZY SETS
MEMBERSHIP
FUNCTION
Crisp
Sets are
special
cases of
Fuzzy
Sets

26. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
DIFFERENCE BETWEEN CRISP AND
FUZZY SET
Membership values allow an element to belong to more than one fuzzy set
In crisp set elements are with extreme values of degree of memebership functions
1 or 0

27. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
TYPICAL REPRESENTAION
OF
FUZZY SETS
.

28. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
written as:
A = µ1/x1 + µ2/x2 + .......... + µn/xn
example : μA = 0.8/x1 + 0.3/x2 + 0.5/x3 + 0.9/x4
Example: “numbers close to 1”
TYPES OF FUZZY SETS

29. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
MEMBERSHIP FUNCTION WITH DISCRETE
MEMEBERSHIP VALUE
A=“Happy Family”

30. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
TYPES OF FUZZY SETS

31. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
MEMBERSHIP FUCTION VALUES WITH
CONTINUOUS MEMBERSHIP VALUES

32. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
CONVEX vs NON-Convex M. F. Distribution

33. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
REPRESENTAION OF FUZZY SETS
four ways for fuzzy membership functions:
 Tabular and list representation (used for finite
sets)
μA = { <x1, 0.8>, <x2, 0.3>, <x3, 0.5>, <x4, 0.9> }
 geometric representation (used for finite sets)
For a set that contains n elements, n-dimensional
Euclidean space is formed and each element may be
represented as a coordinate in that space.

34. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
 Analytic representation
x-5 when 5
μA(x)=
 Graphical representation (most common)
REPRESENTAION OF FUZZY SETS
concept of the fuzzy number
“about six”, “around six”, or
“approximately six”.

35. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
FUZZY MEMBERSHIP FUNCTIONS

36. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
FUZZY MF: Analytical Formulation

37. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
Example:

38. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
FUZZY MF: Analytical Formulation

39. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
Example:

40. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
FUZZY MF: Analytical Formulation

41. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad

42. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
FUZZY MF: Analytical Formulation

43. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad

44. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
FUZZY MF: Analytical Formulation

45. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
FUZZY MF: Analytical Formulation

46. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad

47. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
Difference between crisp and fuzzy
sets

48. Dr. Sonali Gupta
FIC , JCBOSE UST, Faridabad
LINGUISTIC VARIABLES AND VALUES
Using one fuzzy set we can derive or define other fuzzy sets
from it using some mathematical computation and
formulation.