Q1Soft Computing
Question
Q.1. Given two universes X={x1, x2, x3, x4, x5} and Y={y1, y2, y3, y4, y5}, the fuzzy sets A defined on X and fuzzy set B defined on Y are given below. (i) Find the relation R = A x B, where A = {0.4/x1 + 0.7/x2 + 1/x3 + 0.8/x4 + 0.6/x5}, B = {0.2/y1 + 0.6/y2 + 1/y3 + 0.9/y4 + 0.7/y5}. Consider another fuzzy set C defined on the universe V={v1, v2, v3}, C = {0.4/v1 + 1/v2 + 0.8/v3}. (ii) Find P = B x C. (iii) Using max-min composition find R o P.
Answer
Given A = {0.4/x1, 0.7/x2, 1/x3, 0.8/x4, 0.6/x5}, B = {0.2/y1, 0.6/y2, 1/y3, 0.9/y4, 0.7/y5}, and C = {0.4/v1, 1/v2, 0.8/v3}. The fuzzy Cartesian product (relation) R = A x B is computed element-wise as R(xi,yj) = min(A(xi), B(yj)).
| R | y1(0.2) | y2(0.6) | y3(1.0) | y4(0.9) | y5(0.7) |
|---|---|---|---|---|---|
| x1(0.4) | 0.2 | 0.4 | 0.4 | 0.4 | 0.4 |
| x2(0.7) | 0.2 | 0.6 | 0.7 | 0.7 | 0.7 |
| x3(1.0) | 0.2 | 0.6 | 1.0 | 0.9 | 0.7 |
| x4(0.8) | 0.2 | 0.6 | 0.8 | 0.8 | 0.7 |
| x5(0.6) | 0.2 | 0.6 | 0.6 | 0.6 | 0.6 |
Similarly, P = B x C is computed as P(yj,vk) = min(B(yj), C(vk)).
| P | v1(0.4) | v2(1.0) | v3(0.8) |
|---|---|---|---|
| y1(0.2) | 0.2 | 0.2 | 0.2 |
| y2(0.6) | 0.4 | 0.6 | 0.6 |
| y3(1.0) | 0.4 | 1.0 | 0.8 |
| y4(0.9) | 0.4 | 0.9 | 0.8 |
| y5(0.7) | 0.4 | 0.7 | 0.7 |
Finally, R o P is computed using max-min composition: (RoP)(xi,vk) = max over j of min(R(xi,yj), P(yj,vk)). For example, for x1 and v2: min(R(x1,y1),P(y1,v2))=min(0.2,0.2)=0.2; min(R(x1,y2),P(y2,v2))=min(0.4,0.6)=0.4; min(R(x1,y3),P(y3,v2))=min(0.4,1.0)=0.4; min(R(x1,y4),P(y4,v2))=min(0.4,0.9)=0.4; min(R(x1,y5),P(y5,v2))=min(0.4,0.7)=0.4; maximum of these is 0.4.
| RoP | v1 | v2 | v3 |
|---|---|---|---|
| x1 | 0.4 | 0.4 | 0.4 |
| x2 | 0.4 | 0.7 | 0.7 |
| x3 | 0.4 | 1.0 | 0.8 |
| x4 | 0.4 | 0.8 | 0.8 |
| x5 | 0.4 | 0.6 | 0.6 |