## Principle and Fuzzy Relations

### 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 = μ_{A}(x_{1})/x_{1}+ μ_{A}(x_{2})/x_{2}+ ⋅⋅⋅ + μ_{A}(x_{n})/x_{n}

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) = μ_{A}(x_{1})/y_{1}+ μ_{A}(x_{2})/y_{2}+ ⋅⋅⋅ + μ_{A}(x_{n})/y_{n}

where y_{i} = f( x_{i} ), i = 1, ..., n. In other words, the fuzzy set B can be defined through the values of
f(⋅) in x_{1}, ..., x_{n}. If f(⋅) is a many-to-one mapping, then there exist
x_{1}, x_{2} ∈ X, x_{1} ≠ x_{2}, such that f( x_{1} ) = f( x_{2} ) = 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 = x_{1} and x = x_{2}, since f(x) = y^{∗} may result from either x = x_{1} or x = x_{2}. More
generally, we have

μ_{B}(y) = max_{x=f-1(y)}μ_{A}(x)

### Examples

Some examples are shown below:

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

Let: A = 0.1/-2 + 0.4/-1 + 0.8/0 + 0.9/1 + 0.3/2 and f(x) = x^{2}- 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:

#### 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

Let: μ_{A}= bell(x; 1.5, 2, 0.5) and (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 μ_{A}(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 μ_{B}(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 μ_{B}(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 X_{1} × ⋅⋅⋅ X_{n} to a single universe Y
such that f( x_{1}, ..., x_{n}) = y, and there is a fuzzy set A_{i} in each X_{i}, i = 1, ..., n. Since each element in an input vector
( x_{1}, ..., x_{n} ) 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
A

### 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

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