3.3. COMBINING FUZZY SETS 91

These operations are determined by theircharacteristic functions.Thechar-
acteristic functions for intersection, union, and complement are

(A∩B)(x)=

Ω

1 ifx∈Aandx∈B

0 ifx/∈Aorx/∈B

(3.4)

(A∪B)(x)=

Ω

1 ifx∈Aorx∈B

0 ifx/∈Aandx/∈B

(X−A)(x)=

Ω

1 ifx/∈A

0 ifx∈A

For these characteristic functions representing ordinary sets, the following are
satisfied. You will be asked to prove the factsin Equations 3.5 in the exercises.

(A∩B)(x)=A(x)∧B(x)=A(x)B(x) (3.5)

=max{A(x)+B(x)− 1 , 0 }

(A∪B)(x)=A(x)∨B(x)=A(x)+B(x)−A(x)B(x)

=min{A(x)+B(x), 1 }

(X−A)(x)=1−A(x)=(1−A(x)a)

(^1) a

1 −x
1+(a−1)x
fora> 0
Each of the above equations leads to a generalization of AND, OR, or NOT for
fuzzy subsets. By far the most common generalizations are the classical fuzzy
operators
AND: (A∧B)(x)=A(x)∧B(x)(minimum or meet) (3.6)
OR: (A∨B)(x)=A(x)∨B(x)(maximum or join)
NOT: ( ̈A)(x)=1−A(x)(complement or negation)
whereA(x)∧B(x)=min{A(x),B(x)}andA(x)∨B(x)=max{A(x),B(x)}.
The membership functions for these fuzzy sets will be denoted byA∧Bfor min-
imum,A∨Bfor maximum, andA^0 or ̈Afor complement.
Example 3.1SupposeX=[0,1],A(x)=x^2 ,andB(x)=sinπx.Thesets
A∨B,A∧B,andB^0 are shown below.
0
0.2
0.4
0.6
0.8
1
y
0.2 0.4 x 0.6 0.8 1
A∨B(solid),A∧B(dotted)
0
0.2
0.4
0.6
0.8
1
y
0.2 0.4 x 0.6 0.8 1
B(dotted),B^0 (solid)
There is an important relation that exists among these three operations,
known as theDe Morgan Laws.
(A∨B)^0 =A^0 ∧B^0 and(A∧B)^0 =A^0 ∨B^0 (3.7)