Let R and S be two fuzzy relations defined as Then, the resulting relation, T,…
2016
Let R and S be two fuzzy relations defined as
Then, the resulting relation, T, which relates elements of universe of X to elements of universe of Z using max-product composition is given by
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Solution: Compute the relation T = R ∘ S using max-product composition.
Rule: For each x in X and z in Z, T(x,z) = max over y of [ R(x,y) * S(y,z) ].
T(x1,z1) = max(0.7*0.9 = 0.63, 0.5*0.1 = 0.05) = 0.63
T(x1,z2) = max(0.7*0.6 = 0.42, 0.5*0.7 = 0.35) = 0.42
T(x1,z3) = max(0.7*0.2 = 0.14, 0.5*0.5 = 0.25) = 0.25
T(x2,z1) = max(0.8*0.9 = 0.72, 0.4*0.1 = 0.04) = 0.72
T(x2,z2) = max(0.8*0.6 = 0.48, 0.4*0.7 = 0.28) = 0.48
T(x2,z3) = max(0.8*0.2 = 0.16, 0.4*0.5 = 0.20) = 0.20
Final relation T:
Row for x1: [0.63, 0.42, 0.25]
Row for x2: [0.72, 0.48, 0.20]