61.

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For example, in the case of Equation (4.5), if one chooses α=0.1, then the α-cut of G will be G_0.1={0,5}.

4.1.2 Fuzzy set operations

Once a fuzzy subset has been generated, various operations may be carried out to scale the membership grade of each discourse. In general, two operations known as ‘Dilution’ and ‘Concentration’ are used.

Dilution is an operation which, when applied to a fuzzy subset, increases the magnitudes of the membership grades, in such a way that lower membership grades increase more than higher membership grades. Therefore, the difference between all the membership grades will decrease. A typical dilution operation for a fuzzy subset G is

(4.7)

For instance, suppose a fuzzy subset involves only two membership values, namely, 0.9 and 0.4. After the dilution operation using Equation (4.7), these values become 0.948 and 0.632, respectively. Clearly, the difference between the both membership values is reduced.

The concentration operation is the converse of dilution. All the membership grades are decreased when the concentration operation is applied. However, in such cases, higher membership grades decrease much less than do the lower membership grades. Therefore, the concentration operation makes the membership grades in a fuzzy subset more concentrated. A common concentration function is the square operator:

(4.8)

Two membership values, 0.9 and 0.4, are used as an example. After the concentration operation using Equation (4.8), these membership grades become 0.81 and 0.16, respectively. The difference betw een the membership values is increased.

The complement of a fuzzy subset G is denoted by , and is defined as:

(4.9)

For instance, the complement of the fuzzy subset shown in Equation (4.5) is =(0/0)+(0.56/5)+(0.973/27).

In the case of two fuzzy subsets denoted by G and H, the union ()of G and H denoted by μ_GH, is defined as:

(4.10)

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