Unit – 5
Fuzzy Arithmetic
A fuzzy number is a generalization of a regular, real number in the sense that it does not refer to one single value but rather to a connected set of possible values, where each possible value has its own weight between 0 and 1. This weight is called the membership function.
1. Set A must be normal Fuzzy set;
2. of A must be closed interval foe every in (0,1]
3. Support and strong of A must be bounded.
Example 1: It is necessary to compare 2 sensors based upon their detection levels & gain settings the tails of gain settings & sensor detection level with a obtained item being monitored providing typical membership values to reprocess the detection levels for each sensor is given in tales.
Find following membership function:
a) b) c)
Solution:
Gain setting  Detection level of sensor 1(D1)  Detection level sensor 2(D2) 
0  0  0 
10  0.2  0.35 
20  0.35  0.25 
30  0.65  0.8 
40  0.85  0.95 
50  1  1 
= max {}
b) = min {}
c) = 1
Definition: If aA is a fuzzy set defined on universal set X then its fuzzy cardinality is denoted and defined as
Where is membership grade from given fuzzy set A and is a scalar cardinality (total number of elements present in crisp set) of corresponding
Example 1: Find the fuzzy cardinality of A defined as follows:
Solution: Here X= {0,1,2,3,4,5} s and its scalar cardinality are
{0,2,4,5}  
{0,1,2,3,4,5}  
{2,5}  
{0,2,3,4,5}  
{2,4,5}  
{5} 
Thus,
Note: Fuzzy cardinality of Fuzzy set is a fuzzy set
Example 2: Find fuzzy cardinality of fuzzy set A and B whose membership function are given as follows , B(x)= where x
Solution:
X  0  1  2  3  4 
,  0  0.5  0.67  0.75  0.8 
B(x)=  1  0.9  0.8  0.7  0.6 
{0,1,2,3,4}  
{1,2,3,4}  
{2,3,4}  
{3,4}  
{4} 
{0}  
{0,1}  
{0,1,2}  
{0,1,2,3}  
{0,1,2,3,4} 
Thus,
Thus,
Moving from intervals we can define arithmetic on fuzzy numbers based on principles of interval Arithmetic.
Let A and B denote fuzzy numbers and let * denote any of the four basic arithmetic operations. Then, we define a fuzzy set on R, A*B by defining its alphacut as:
Arithmetic Operations on Intervals: Arithmetic operations on fuzzy intervals satisfy following useful properties:
Properties:
Examples Illustrating intervalvalued arithmetic operations:
Applying the extension principle to arithmetic operations, we have
Fuzzy Addition:
Fuzzy Subtraction
Fuzzy Multiplication
Fuzzy Division
Example 1:
Their
Step 1: Find and
Step 2: Use decomposition theorem which is given as and arithmetic operation on closed interval
Step 3: Find membership function.
Example 2: The membership function of two fuzzy numbers A and B are given as follows
A(x)  =  1 < x 
B(x)  =  
=  =  
= 0  = 0  otherwise 
Find A+B and AB
Solution:
Step 1: Find and
For 1 < x
By def. of ,
A(x)
For
= [21, 32 for all
For
By def. of cut,
B(x)
For 3 < x
B(x)
= [2+1, 52 for all
Step 2: By decomposition theorem & arithmetic operation on closed interval we find as follows.
Step 3: Now find membership function
Let 4
 Let 84 
Thus, (A+B) (x) = ; 0 < x
= 4
= 0; Otherwise
To find AB
Step 4: By decomposition theorem and arithmetic operation on closed interval we find follows
= [
= [4 for all
Step 5: Now find membership function to define fuzzy numbers AB
Let 4
 Let 24

Thus,
Algorithm:
Step 1: Find
Step 2: Use decomposition theorem which is given as
Let
A+X = B
Taking alpha cut on both side
Step 3: Find membership function
Example 1: The membership function of two fuzzy numbers A and B are
A(x)  = x1  1 < x 
B(x)  =  
=  =  
= 0  = 0  otherwise 
Solve A+X =B
Solution:
Let
A+X = B
Taking alpha cut on both side
Check
0.1  12.8  56.9 
0.6  26.8  46.4 
0.8  32.4  42.2 
Thus, hence solution exists.
Let
 Let

Example 2: The membership function of two fuzzy numbers A and B are
A(x)  = x3  3 < x 
B(x)  =  
=  =  
= 0  = 0  otherwise 
Solve A.X =B
Solution:
Let
A.X = B
Taking alpha cut on both side
Check
0.3  4.3636  6.0426 
0.7  4.7568  5.4884 
0.9  4.9231  5.1707 
Thus, hence solution exists.
Let
 Let

Reference