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 = RS = μT(x, z) = \begin{matrix} \vee\\ y \in Y \end{matrix} ( μR(x, y) ∨ μS(y, z) )

= \begin{matrix} max \\ y \in Y \end{matrix} ( min ( μR(x, y) ∨ μS(y, z) ) )

Fuzzy Max–Product composition is defined as:

T = RS = μT(x, z) = \begin{matrix} \vee\\ y \in Y \end{matrix} ( μR(x, y) ∙ μS(y, z) )

= \begin{matrix} max \\ y \in Y \end{matrix} ( ( μR(x, y) ∙ μS(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 = {x1, x2}, Y = {y1, y2},  and Z = {z1, z2, z3}. 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:

Max-min composition is defined as,

T = RS = μT(x, z) = \begin{matrix} \vee\\ y \in Y \end{matrix} ( μR(x, y) ∨ μS(y, z) )

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

μT(x1, z1) = max ( min( μR(x1, y1), μS(y1, z1)), min( μR(x1, y2), μS(y2, z1)) )

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

μT(x1, z2) = max ( min( μR(x1, y1), μS(y1, z2)), min( μR(x1, y2), μS(y2, z2)) )

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

μT(x1, z3) = max ( min( μR(x1, y1), μS(y1, z3)), min( μR(x1, y2), μS(y2, z3)) )

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

μT(x2, z1) = max ( min( μR(x2, y1), μS(y1, z1)), min( μR(x2, y2), μS(y2, z1)) )

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

μT(x2, z2) = max ( min( μR(x2, y1), μS(y1, z2)), min( μR(x2, y2), μS(y2, z2)) )

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

μT(x2, z3) = max ( min( μR(x2, y1), μS(y1, z3)), min( μR(x2, y2), μS(y2, z3)) )

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

Max-Product Composition:

T = RS = μT(x, z) = \begin{matrix} \vee\\ y \in Y \end{matrix} ( μR(x, y) ∙ μS(y, z) )

= \begin{matrix} max \\ y \in Y \end{matrix} ( ( μR(x, y) ∙ μS(y, z) ) )

μT(x1, z1) = max ( (μR(x1, y1) × μS(y1, z1)), ( μR(x1, y2) × μS(y2, z1)) )

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

μT(x1, z2) = max ( ( μR(x1, y1) × μS(y1, z2)), ( μR(x1, y2) × μS(y2, z2)) )

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

μT(x1, z3) = max ( ( μR(x1, y1) × μS(y1, z3)), ( μR(x1, y2) × μS(y2, z3)) )

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

μT(x2, z1) = max ( ( μR(x2, y1) × μS(y1, z1)), ( μR(x2, y2) × μS(y2, z1)) )

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

μT(x2, z2) = max ( ( μR(x2, y1) × μS(y1, z2)), ( μR(x2, y2) × μS(y2, z2)) )

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

μT(x2, z3) = max ( ( μR(x2, y1) × μS(y1, z3)), ( μR(x2, y2) × μS(y2, z3)) )

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