Extension Principle

The extension principle is a basic concept of fuzzy set theory that provides a general procedure for extending crisp domains of mathematical expressions to fuzzy domains. This procedure generalizes a common point-to-point mapping of a fuction f(⋅) to a mapping between fuzzy sets. More specifically, suppose that f is a fuction from X to Y and A is a fuzzy set on X defined as

  A = μA(x1)/x1 + μA(x2)/x2 + ⋅⋅⋅ + μA(xn)/xn

Then the extension principle states that the image of fuzzy set A under the mapping f(⋅) can be expressed as a fuzzy set B,

  B = f(A) = μA(x1)/y1 + μA(x2)/y2 + ⋅⋅⋅ + μA(xn)/yn

where yi = f( xi ), i = 1, ..., n. In other words, the fuzzy set B can be defined through the values of f(⋅) in x1, ..., xn. If f(⋅) is a many-to-one mapping, then there exist x1, x2 ∈ X, x1 ≠ x2, such that f( x1 ) = f( x2 ) = y , y ∈ Y. In this case, the membership grade of B at y = y is the maximum of the membership grades of A at x = x1 and x = x2, since f(x) = y may result from either x = x1 or x = x2. More generally, we have

  μB(y) = maxx=f-1(y)μA(x)


Some examples are shown below:

Application of the extension principle to fuzzy set with discrete universes:

    A = 0.1/-2 + 0.4/-1 + 0.8/0 + 0.9/1 + 0.3/2
    f(x) = x2 - 3

Upon applying the extension principle, we have

  B = 0.1/1 + 0.4/-2 + 0.8/-3 + 0.9/-2 + 0.3/1
    = 0.8/-3 +  (0.4 ∨ 0.9)/-2 + (0.1 ∨ 0.3)/1
    = 0.8/-3 + 0.9/-2 + 0.3/1

where ∨ represents max. This example is illustrated in the figure below:

Extension Principle

Application of the extension principle to fuzzy sets with continuous universe

For a fuzzy set with a continuous universe of discourse X, an analogous procedure applies

    μA = bell(x; 1.5, 2, 0.5)
              (x -1)2 -1,  if x ≤ 0
    f(a) = 
              x,           if x ≤ 0 

Figure 2(a) is the plot of y = f(x); Figure 2(c) is μA(x), the MF of A. After employing the extension principle, we obtain a fuzzy set B; its MF is shown in Figure 2(b), where the plot of μB(y) is rotated 90 degrees for easy viewing. Since f(x) is a many-to-one mapping for x ∈ [-1, 2], the max operator is used to obtain the membership grades of b when y ∈ [0, 1]. This causes discontinuities of μB(y) and y = 0 and -1.

Now we consider a more general situation. Suppose that f is a mapping from a n-dimensional product space X1 × ⋅⋅⋅ Xn to a single universe Y such that f( x1, ..., xn) = y, and there is a fuzzy set Ai in each Xi, i = 1, ..., n. Since each element in an input vector ( x1, ..., xn ) occurs simultaneously, this implies an AND operation. Therefore, the membership grade of fuzzy set B induced by the mapping f should be the minimum of the membership grades of the constituent fuzzy set Ai, i = 1, ..., n. With this understanding, we give a complete formal definition of the extension principle:

Fuzzy Relations

Binary fuzzy relations [4, 6] are fuzzy set in X × Y which map each element in X × Y to a membership grade between 0 and 1. In particular, unary fuzzy relations are fuzzy sets with one-dimensional MFs; binary fuzzy relations are fuzzy sets with two-dimensional MFs, and so on. Applications of fuzzy relations include areas such as fuzzy control and decision making. Here we restrict our attention to binary fuzzy relations; a generalization to n-ary relations is straightforward.

Binary Fuzzy Relation