🔗 Min-Max Composition in Fuzzy Logic

📌 What is Min-Max Composition?

Min-Max Composition is a method to combine two fuzzy relations to form a third fuzzy relation, based on relational reasoning.

This is especially used when we have multiple fuzzy sets or stages in a fuzzy inference chain, like A → B → C, and we want to calculate the indirect relation between A and C.

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🧠 Why is it Important?

• It forms the basis of fuzzy inference and rule chaining.

• Helps in decision-making systems and pattern recognition models.

• Widely used in Fuzzy Inference Systems (FIS) when combining antecedent and consequent fuzzy rules.

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🔢 Mathematical Definition:

Given two fuzzy relations:

• R_1: Relation from X to Y,

• R_2: Relation from Y to Z

Then, the Min-Max Composition R = R_1 \circ R_2 is defined as:

\mu_R(x, z) = \max_{y} \left[ \min\left( \mu_{R_1}(x, y), \mu_{R_2}(y, z) \right) \right]

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📊 Matrix Representation Example

Let’s say we have:

• R1 (X → Y):

R_1 = \begin{bmatrix} 0.3 & 0.8 \ 0.7 & 0.4 \end{bmatrix}

• R2 (Y → Z):

R_2 = \begin{bmatrix} 0.5 & 0.6 \ 0.9 & 0.2 \end{bmatrix}

Now we calculate R3 = R1 ∘ R2 (X → Z) using Min-Max Composition:

Step 1: Calculate Each Element of Result Matrix R3

• R_3(1,1) = \max[\min(0.3, 0.5), \min(0.8, 0.9)] = \max(0.3, 0.8) = 0.8

• R_3(1,2) = \max[\min(0.3, 0.6), \min(0.8, 0.2)] = \max(0.3, 0.2) = 0.3

• R_3(2,1) = \max[\min(0.7, 0.5), \min(0.4, 0.9)] = \max(0.5, 0.4) = 0.5

• R_3(2,2) = \max[\min(0.7, 0.6), \min(0.4, 0.2)] = \max(0.6, 0.2) = 0.6

Final Composed Matrix:

R_3 = \begin{bmatrix} 0.8 & 0.3 \ 0.5 & 0.6 \end{bmatrix}

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📝 Exam-Ready Answer Format:

Min-Max Composition is a technique to combine two fuzzy relations. It is defined as the maximum of minimum values between corresponding elements of two relations. It is used in fuzzy inference systems, fuzzy relational databases, and fuzzy decision-making. If R1 and R2 are fuzzy relations, then the composition R = R1 ∘ R2 is computed as \mu_R(x,z) = \max_y[\min(\mu_{R1}(x,y), \mu_{R2}(y,z))].

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🔍 Real-World Applications of Min-Max Composition:

Domain	Use Case
Healthcare	Relation between symptoms → diseases → treatments
Expert Systems	Rule chaining in decision-making logic
NLP	Word-sentence-document relevance chains
Fuzzy Control	Combining input-output fuzzy rules across multiple stages

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🎯 Mnemonic for Quick Recall: “Min then Max”

• For each pair (x, z):

1. Take min across all paths (via y)

2. Then take max of all those mins.