FUZZY SETS
AND
FUZZY LOGIC
Theory and
Applications
PART 2
Fuzzy sets vs
crisp sets
1. Properties of α-cuts
2. Fuzzy set representations
3. Extension principle
Properties of α-cuts
• Theorem 2.1
Let A, B  F(X). Then, the following
properties hold for all α, β 
[0, 1]:
(i)   A A;







implies
A

A
and
A

A;
(ii)
(iii)  ( A  B) A B and  ( A  B) A B;






(iv) ( A  B) A B and ( A  B) A B;
(v)  ( A ) (1 )A
Properties of α-cuts
Properties of α-cuts
• Theorem 2.2
Let Ai  F(X) for all i  I, where I is an
index set. Then,
(vi)

iI
(vii) 
iI

Ai  ( Ai ) and   Ai  ( Ai );
iI

iI

Ai  ( Ai ) and 
iI
iI
iI


Ai  ( Ai ).
iI
Properties of α-cuts
1
• Let Ai ( x)  1  , I  N ,   1, X arbitrary
i
1
Ai  
  1 Ai  
iI
But ( Ai )(x)  sup Ai ( x)  1
iI
iI
  Ai  X
iI
1 ( Ai )  X
iI
  1 Ai 1 ( Ai )
iI
iI
Properties of α-cuts
• Theorem 2.3
Let A, B  F(X). Then, for all α


(viii) A  B iff A B;
A  B iff


A B;
(ix) A  B iff  A B;
A  B iff


A B.
[0, 1],
Properties of α-cuts
• Theorem 2.4
For any A  F(X), the following properties
hold:
(x)

A

 

(xi)
A

 

A 

A



 

A;


 
A;.
Fuzzy set representations
• Theorem 2.5 (First Decomposition Theorem)
For every A  F(X),
A

 A,

[ 0,1]
where αA is defined by (2.1), and ∪
denotes the standard fuzzy union.
Fuzzy set representations
Fuzzy set representations
• Theorem 2.6 (Second Decomposition Theorem)
For every A  F(X),
A




A,
[ 0,1]
where α+A denotes a special fuzzy set
defined by

A
(
x
)



A( x)

and ∪ denotes the standard fuzzy union.
Fuzzy set representations
• Theorem 2.7 (Third Decomposition Theorem)
For every A  F(X),
A

 A,

 ( A)
where Λ(A) is the level of A, αA is defined
by (2.1), and ∪denotes the standard fuzzy
union.
Fuzzy set representations
Extension principle
Extension principle
• Extension principle.
Any given function f : X→Y induces two
functions,
f : F( X )  F(Y ),
f
1
: F(Y )  F( X ),
Extension principle
which are defined by
[ f ( A)](y)  sup A( x)
x| y  f ( x )
for all A  F(X) and
1
[ f ( B)](x)  B( f ( x))
for all B  F(Y).
Extension principle
Extension principle
Extension principle
• Theorem 2.8
Let f : X→Y be an arbitrary crisp function.
Then, for any Ai  F(X) and any Bi  F(Y),
i  I, the following properties of functions
obtained by the extension principle hold:
Extension principle
Extension principle
• Theorem 2.9
Let f : X→Y be an arbitrary crisp function.
Then, for any Ai  F(X) and all α [0, 1]
the following properties of fuzzified by the
extension principle hold:
Extension principle
• Let X  N ,   1,
 a, x  10
f ( x)  
 b, x  10
1
A( x)  1 
x
Extension principle
Extension principle
• [ f ( A)](a )  sup A( x)  0.9
x| x  a
[ f ( A)](b)  sup A( x)  1
x| x  b
1 [ f ( A)]  0 a  1 b
But 1A    f ( )    0 a  0 b
 f ( A)  [ f ( A)].
1
1
Extension principle
• Theorem 2.10
Let f : X→Y be an arbitrary crisp function.
Then, for any Ai  F(X), f fuzzified by the
extension principle satisfies the equation:
f ( A) 
f (


[ 0,1]

A)
Exercise 2
• 2.4
• 2.8
• 2.11