Let R and S be two fuzzy relations defined as R (rows x₁, x₂; columns y₁, y₂):
Let R and S be two fuzzy relations defined as
R (rows x₁, x₂; columns y₁, y₂):
x₁: [0.7 0.5]
x₂: [0.8 0.4]
and S (rows y₁, y₂; columns z₁, z₂, z₃):
y₁: [0.9 0.6 0.2]
y₂: [0.1 0.7 0.5]
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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Correct answer: (C) T (columns z₁, z₂, z₃): x₁: [0.63 0.42 0.35], x₂: [0.72 0.48 0.20]
Explanation
Max-product composition: T(x, z) = max over y of R(x, y) × S(y, z). T(x₁, z₁) = max(0.7 × 0.9, 0.5 × 0.1) = 0.63; T(x₁, z₂) = max(0.42, 0.35) = 0.42; T(x₁, z₃) = max(0.14, 0.25) = 0.35. T(x₂, z₁) = max(0.72, 0.04) = 0.72; T(x₂, z₂) = max(0.48, 0.28) = 0.48; T(x₂, z₃) = max(0.16, 0.20) = 0.20. So T = x₁: [0.63 0.42 0.35], x₂: [0.72 0.48 0.20].