Composition of fuzzy relation is defined over two fuzzy relations.

### Fuzzy composition

**Fuzzy composition **can be defined just as it is for crisp (binary) relations. Suppose R is a fuzzy relation on X × Y, S is a fuzzy relation on Y × Z, and T is a fuzzy relation on X × Z; then,

**Fuzzy Max–Min composition **is defined as:

T = R ∘ S = μ_{T}(x, z) = \begin{matrix} \vee\\ y \in Y \end{matrix} ( μ_{R}(x, y) ∨ μ_{S}(y, z) )

= \begin{matrix} max \\ y \in Y \end{matrix} ( min ( μ_{R}(x, y) ∨ μ_{S}(y, z) ) )

**Fuzzy Max–Product composition **is defined as:

T = R ∘ S = μ_{T}(x, z) = \begin{matrix} \vee\\ y \in Y \end{matrix} ( μ_{R}(x, y) ∙ μ_{S}(y, z) )

= \begin{matrix} max \\ y \in Y \end{matrix} ( ( μ_{R}(x, y) ∙ μ_{S}(y, z) ) )

Let us try to understand it with the help of example. The fuzzy max-min approach is identical that of the crisp max-min composition.

**Example:**

X = {x_{1}, x_{2}}, Y = {y_{1}, y_{2}}, and Z = {z_{1}, z_{2}, z_{3}}*. *Consider the following fuzzy relations:

Find the resulting relation, T which relates elements of universe X to elements of universe Z, i.e., defined on Cartesian space X × Z

- Using
**Max–Min**composition and - Using
**Max-Product**composition

**Solution:**

So ultimately, we have to find the elements of matrix,

Max-min composition is defined as,

T = R ∘ S = μ_{T}(x, z) = \begin{matrix} \vee\\ y \in Y \end{matrix} ( μ_{R}(x, y) ∨ μ_{S}(y, z) )

From the given relations \bar{R} and \bar{S},

μ_{T}(x_{1}, z_{1}) = max ( min( μ_{R}(x_{1}, y_{1}), μ_{S}(y_{1}, z_{1})), min( μ_{R}(x_{1}, y_{2}), μ_{S}(y_{2}, z_{1})) )

= max(min(0.7, 0.8), min(0.6, 0.1)) = max(0.7, 0.1) = 0.7

μ_{T}(x_{1}, z_{2}) = max ( min( μ_{R}(x_{1}, y_{1}), μ_{S}(y_{1}, z_{2})), min( μ_{R}(x_{1}, y_{2}), μ_{S}(y_{2}, z_{2})) )

= max(min(0.7, 0.5), min(0.6, 0.6)) = max(0.5, 0.6) = 0.6

μ_{T}(x_{1}, z_{3}) = max ( min( μ_{R}(x_{1}, y_{1}), μ_{S}(y_{1}, z_{3})), min( μ_{R}(x_{1}, y_{2}), μ_{S}(y_{2}, z_{3})) )

= max(min(0.7, 0.4), min(0.6, 0.7)) = max(0.4, 0.6) = 0.6

μ_{T}(x_{2}, z_{1}) = max ( min( μ_{R}(x_{2}, y_{1}), μ_{S}(y_{1}, z_{1})), min( μ_{R}(x_{2}, y_{2}), μ_{S}(y_{2}, z_{1})) )

= max(min(0.8, 0.8), min(0.3, 0.1)) = max(0.8, 0.1) = 0.8

μ_{T}(x_{2}, z_{2}) = max ( min( μ_{R}(x_{2}, y_{1}), μ_{S}(y_{1}, z_{2})), min( μ_{R}(x_{2}, y_{2}), μ_{S}(y_{2}, z_{2})) )

= max(min(0.8, 0.5), min(0.3, 0.6)) = max(0.5, 0.3) = 0.5

μ_{T}(x_{2}, z_{3}) = max ( min( μ_{R}(x_{2}, y_{1}), μ_{S}(y_{1}, z_{3})), min( μ_{R}(x_{2}, y_{2}), μ_{S}(y_{2}, z_{3})) )

= max(min(0.8, 0.4), min(0.3, 0.7)) = max(0.4, 0.3) = 0.4

T = R ∘ S = μ_{T}(x, z) = \begin{matrix} \vee\\ y \in Y \end{matrix} ( μ_{R}(x, y) ∙ μ_{S}(y, z) )

= \begin{matrix} max \\ y \in Y \end{matrix} ( ( μ_{R}(x, y) ∙ μ_{S}(y, z) ) )

μ_{T}(x_{1}, z_{1}) = max ( (μ_{R}(x_{1}, y_{1}) × μ_{S}(y_{1}, z_{1})), ( μ_{R}(x_{1}, y_{2}) × μ_{S}(y_{2}, z_{1})) )

= max((0.7 × 0.8), (0.6 × 0.1)) = max(0.56, 0.06) = 0.56

μ_{T}(x_{1}, z_{2}) = max ( ( μ_{R}(x_{1}, y_{1}) × μ_{S}(y_{1}, z_{2})), ( μ_{R}(x_{1}, y_{2}) × μ_{S}(y_{2}, z_{2})) )

= max( (0.7 × 0.5), (0.6 × 0.6)) = max(0.35, 0.36) = 0.36

μ_{T}(x_{1}, z_{3}) = max ( ( μ_{R}(x_{1}, y_{1}) × μ_{S}(y_{1}, z_{3})), ( μ_{R}(x_{1}, y_{2}) × μ_{S}(y_{2}, z_{3})) )

= max((0.7 × 0.4), (0.6 × 0.7)) = max(0.28, 0.42) = 0.42

μ_{T}(x_{2}, z_{1}) = max ( ( μ_{R}(x_{2}, y_{1}) × μ_{S}(y_{1}, z_{1})), ( μ_{R}(x_{2}, y_{2}) × μ_{S}(y_{2}, z_{1})) )

= max((0.8 × 0.8), min(0.3 × 0.1)) = max(0.64, 0.03) = 0.64

μ_{T}(x_{2}, z_{2}) = max ( ( μ_{R}(x_{2}, y_{1}) × μ_{S}(y_{1}, z_{2})), ( μ_{R}(x_{2}, y_{2}) × μ_{S}(y_{2}, z_{2})) )

= max((0.8 × 0.5), (0.3 × 0.6)) = max(0.4, 0.18) = 0.40

μ_{T}(x_{2}, z_{3}) = max ( ( μ_{R}(x_{2}, y_{1}) × μ_{S}(y_{1}, z_{3})), ( μ_{R}(x_{2}, y_{2}) × μ_{S}(y_{2}, z_{3})) )

= max((0.8 × 0.4), (0.3 × 0.7)) = max(0.32, 0.21) = 0.32

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## Test Your Knowledge:

Let two sets P = {P_{1}, P_{2}, P_{3}, P_{4}} and D = {D_{1}, D_{2}, D_{3}, D_{4}} represent a set of variety of ** paddy plants **and a set of

**. In addition to these, also consider another set S = {S**

*plant diseases*_{1}, S

_{2}, S

_{3}, S

_{4}} be the

**common symptoms**of the diseases. Let, R be a relation on P x D, representing which plant is susceptible to which diseases, then R can be stated as,

Also, consider T be the another relation on D x S, which is given by

Find the association of plants with the different symptoms of the disease using ** max-min composition**.

**Please post your answer / query / feedback in comment section below !**

Codecrucks is the best place for learning. Thanks Dr. Mahesh Goyani for this great tutorials. 👍

Noted with thanks Komal

Very systematic and easy way to learn!!!!

Thanks Nazeera

Thanks to this blog of fuzzy composition. Its too much helpful and specifically to ME students. Explained in minutely detailed fashion.

Noted with thanks