Compute the value of adding the following two fuzzy integers : A = {(0.3, 1),…
2016
Compute the value of adding the following two fuzzy integers :
A = {(0.3, 1), (0.6, 2), (1, 3), (0.7, 4), (0.2, 5)}
B = {(0.5, 11), (1, 12), (0.5, 13)}
Where fuzzy addition is defined as
\(\mu_{A+B} (z) = max_{x+y=z} (min (\mu_A(x), \mu_b(x)))\)
Then, f (A + B) is equal to
- A.
{(0.5, 12), (0.6, 13), (1, 14), (0.7, 15), (0.7, 16), (1, 17), (1, 18)}
- B.
{(0.5, 12), (0.6, 13), (1, 14), (1, 15), (1, 16), (1, 17), (1, 18)}
- C.
{(0.3, 12), (0.5, 13), (0.5, 14), (1, 15), (0.7, 16), (0.5, 17), (0.2, 18)}
- D.
{(0.3, 12), (0.5, 13), (0.6, 14), (1, 15), (0.7, 16), (0.5, 17), (0.2, 18)}
Attempted by 25 students.
Show answer & explanation
Correct answer: D
Solution:
Use μ_{A+B}(z) = max_{x+y=z} min(μ_A(x), μ_B(y)). Compute for z = 12,...,18.
z = 12: only pair is (1,11) → min(0.3,0.5) = 0.3. So μ_{A+B}(12) = 0.3.
z = 13: pairs are (1,12) → min(0.3,1)=0.3 and (2,11) → min(0.6,0.5)=0.5. Max{0.3,0.5} = 0.5. So μ_{A+B}(13) = 0.5.
z = 14: pairs are (1,13) → min(0.3,0.5)=0.3, (2,12) → min(0.6,1)=0.6, (3,11) → min(1,0.5)=0.5. Max{0.3,0.6,0.5} = 0.6. So μ_{A+B}(14) = 0.6.
z = 15: pairs are (2,13) → min(0.6,0.5)=0.5, (3,12) → min(1,1)=1, (4,11) → min(0.7,0.5)=0.5. Max{0.5,1,0.5} = 1. So μ_{A+B}(15) = 1.
z = 16: pairs are (3,13) → min(1,0.5)=0.5, (4,12) → min(0.7,1)=0.7, (5,11) → min(0.2,0.5)=0.2. Max{0.5,0.7,0.2} = 0.7. So μ_{A+B}(16) = 0.7.
z = 17: pairs are (4,13) → min(0.7,0.5)=0.5 and (5,12) → min(0.2,1)=0.2. Max{0.5,0.2} = 0.5. So μ_{A+B}(17) = 0.5.
z = 18: only pair is (5,13) → min(0.2,0.5) = 0.2. So μ_{A+B}(18) = 0.2.
Final result: {(0.3, 12), (0.5, 13), (0.6, 14), (1, 15), (0.7, 16), (0.5, 17), (0.2, 18)}