Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012 Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh Reference Books: 1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004. 2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005. 3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012. Please comment and feel free to ask anything related. Thanks!

1. EE-646
Lecture-3
Some other Characteristics of
Fuzzy Sets

2. 5-Sep-12 2

3. 1. Support of a Fuzzy Set
is the set of all points x in X such that µA(x) > 0
It is given by:
Supp (A) or S(A)
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{ }: ( ) 0,Ax x x Xµ= > ∀ ∈

4. 2. Core or Nucleus
of a fuzzy set A is the set of all points x in X
such that µA(x) = 1
It is given by:
NUC (A) or Core(A)
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{ }: ( ) 1,Ax x x Xµ= = ∀ ∈

5. 3. Normality or Normal Fuzzy Set
A fuzzy set is normal if its core is non-empty.
In other words, there exists at least one point
x in X such that µA(x) = 1
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6. 5-Sep-12 6
Normal Fuzzy Set
x
Subnormal
Fuzzy Set

7. 4. Height of a Fuzzy Set
It is the largest membership value of an
element in A and is defined as:
Height(A) or H(A)
Sup stands for supremum which means max.
Height for a normal set is 1.
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{ }Sup ( )A
x X
xµ
∈
=

8. 5. Crossover Points
is(are) point(s) at which µA(x) = 0.5
There can be at most two crossover points for a
convex fuzzy set
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9. 5-Sep-12 9
x
x1 x2

10. 6. Bandwidth (or width)
For a normal and convex fuzzy set the BW or
width is defined as the distance between two
unique crossover points
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11. 5-Sep-12 11
x
x1 x2
BW (A) = |x1 – x2|

12. 7. Fuzzy Singleton set
A fuzzy set whose support is a single point in X
with µA(x) = 1 is called a fuzzy singleton set
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x
1.0
Nucleus & Support

13. 6. α - cut set
An α-cut or α-level set of a fuzzy set A ⊆ X is a
crisp set Aα ⊆ X, where α ϵ [0, 1], such that:
A strong α-cut set is denoted by Aα+ & is given
by:
This concept is used in defuzzification
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{ }( ) ( )AA
x xαµ µ α= ≥
{ }( ) ( )AA
x xαµ µ α+= >

14. α - cut set example
Consider the discrete fuzzy set, defined on the
defined on universe X = {a, b, c, d, e, f }
Find A1, A0.9, A0.6, A0.3, A0, A0+
Also verify that Core (A) = A1
& Support (A) = A0+
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1 0.9 0.6 0.3 0.01 0
+ + + +A
a b c d e f
 
+ 
 

15. Continuous α-cuts
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α
α

16. Order of Magnitude
Core (A) < Aα1 <Aα2< Support (A)
Where α1 > α2
Also note that all of the above discussed sets
are crisp sets
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17. 9. Convexity
Convex Linear Combination
Convexity of a function
By varying we move from x1 to x2. It gives the
convexity of the system.
A fuzzy set A is convex iff for
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( ) ( ) ( )1 2 1 2(1 ) (1 ) , [0, 1]f x x f x f xλ λ λ λ λ+ − ≥ + − ∈
( ) ( )
( ) ( ) ( )
1 2 1 2
1 2 1 2
(1 ) (1 )
(1 ) min ,
A A
A A A
x x x x
x x x x
µ λ λ µ λ λ
µ λ λ µ µ
+ − ≥ + −
 + − ≥  
1 2, and [0, 1]x x X λ∈ ∈

18. 9. Convexity...contd
Alternatively, A is convex if all of its α-cut level
sets are convex.
Another Definition:
If for any two points on a curve, the line
drawn between them is completely inside the
curve and does not cut it at any other point,
then the curve is said to be convex.
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19. Illustration
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20. Illustration
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