Unit – 5
Fuzzy Arithmetic
Q1) 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)
A1)
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-
Q2) Find the fuzzy cardinality of A defined as follows:
A2)
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 s
Q3) Find fuzzy cardinality of fuzzy set A and B whose membership function are given as follows 
, B(x)=
where x
A3)
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,
Q4)
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.
Q5) 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 A-B
A5)
Step 1: Find 
and 
For -1 < x 
By def. of 
,
A(x)
For 
= [2
-1, 3-2
for all 
For 
By def. of 
cut,
B(x)
For 3 < x 
B(x)
= [2
+1, 5-2
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 8-4
|
Thus, (A+B) (x) = 
; 0 < x 
= 
4
= 0; Otherwise
To find A-B
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 A-B
Let 4
| Let 2-4
|
Thus,
Q6) The membership function of two fuzzy numbers A and B are
A(x) | = x-1 | 1 < x |
B(x) | = |
|
= |
| = |
| ||
= 0 |
| = 0 | otherwise |
Solve A+X =B
A6)
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
|
Q7) The membership function of two fuzzy numbers A and B are
A(x) | = x-3 | 3 < x |
B(x) | = |
|
= |
| = |
| ||
= 0 |
| = 0 | otherwise |
Solve A.X =B
A7)
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
|
