Q4) b) Explain with Example Min-Max Composition.

✅ Definition:

Min-Max Composition is a method to combine two fuzzy relations (R1 and R2) to get a new fuzzy relation (R3), using min of pairs and max across all pairs.

Formula:

\mu_{R3}(x,z) = \max_y \left[\min\left(\mu_{R1}(x,y), \mu_{R2}(y,z)\right)\right]

• Min: for each pair (x–y and y–z)

• Max: selects the strongest (maximum) connection via all y’s

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📊 Example:

Let:

R1 (X→Y) =

\begin{bmatrix} 0.3 & 0.8 \ 0.7 & 0.4 \end{bmatrix}

R2 (Y→Z) =

\begin{bmatrix} 0.5 & 0.6 \ 0.9 & 0.2 \end{bmatrix}

🔢 Calculation of R3 (X→Z):

• R3(1,1) = max[min(0.3, 0.5), min(0.8, 0.9)] = max(0.3, 0.8) = 0.8

• R3(1,2) = max[min(0.3, 0.6), min(0.8, 0.2)] = max(0.3, 0.2) = 0.3

• R3(2,1) = max[min(0.7, 0.5), min(0.4, 0.9)] = max(0.5, 0.4) = 0.5

• R3(2,2) = max[min(0.7, 0.6), min(0.4, 0.2)] = max(0.6, 0.2) = 0.6

✅ Final Result:

R3 (X→Z) =

\begin{bmatrix} 0.8 & 0.3 \ 0.5 & 0.6 \end{bmatrix}

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Conclusion:

Min-Max Composition is used in fuzzy inference systems and decision chains where indirect relations are derived. It is useful in fuzzy databases, rule chaining, and control systems.