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There are two common forms of composition operation in Fuzzy Theory:

  • max–min composition
  • max–product composition

Let R be a relation that relates elements from universe X to universe Y, and let S be a relation that relates elements from universe Y to universe Z.

Relation R and S are as follows

$R = \begin{array}{c c} & \begin{array}{c c c} y_1 & y_2 \\ \end{array}\\ \begin{array}{c c c} x_1 \\ x_2 \\ x_3 \end{array} & \left[\begin{array}{c c c} 0.1 & 0.3 \\ 0.4 & 0.2 \\ 0.8 & 0.6 \end{array}\right] \end{array}$ $S = \begin{array}{c c} & \begin{array}{c c c} z_1 & z_2 & z_3 \\ \end{array}\\ \begin{array}{c c c} y_1 \\ y_2 \\ \end{array} & \left[\begin{array}{c c c} 0.1 & 0.3 & 0.2 \\ 0.6 & 0.4 & 0.5 \\ \end{array}\right] \end{array}$

Now relation between $x_1$ and $z_1$ can be calculated as

max–min composition= max[min(0.1, 0.1), min(0.3, 0.6)] = 0.3

max–product composition = max[(0.1*0.1), (0.3*0.6)] = 0.18

In Crisp Set it is easy to get the intuition i.e. if $x_1$ related $y_1$ and if $y_1$ is related to $z_1$ then with transitive property we can say $x_1$ is related to $z_1$. But in the case of Fuzzy Set Theory what these operations intuitively say. What does it mean to say, $x_1$ is related to $z_1$ with 0.3 in the case of max–min composition and with 0.18 in the case of max–product composition intuitively.

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