The application of fuzzy mathematics in real estate appraisal is particularly important.

There are m fuzzy subsets (m models) on the universe U={ x 1, x2, …, xn}, which constitutes the standard model base. The recognized object is also a fuzzy set, which is closest to it? This is a problem of identifying standard fuzzy sets with fuzzy sets. Therefore, this involves the closeness of two fuzzy sets.

1, intimacy

Firstly, the inner product and outer product of fuzzy vectors are extended to infinite universe u, and the simple properties of inner product and outer product are also true for fuzzy sets on infinite universe u.

According to the properties of inner product and outer product of fuzzy sets, it can be seen that the closeness between two fuzzy sets can not be completely described by using inner product or outer product alone. The inner product and outer product of fuzzy sets can only partially represent the closeness of two fuzzy sets. Now further illustrate this point intuitively. The greater the ordinate (membership degree) of the intersection of two fuzzy sets shown in figure 1, the closer it is, and the inner product just represents the ordinate (membership degree μ) of the intersection of fuzzy sets. The smaller the ordinate (membership μ) of the intersection of two fuzzy sets shown in Figure 2, the closer it is, and the outer product ⊙ = just illustrates this point.

To sum up, the greater the inner product, the closer the fuzzy set is; The smaller the outer product, the closer the fuzzy set is. Therefore, it is more appropriate to describe the closeness of two fuzzy sets with the "closeness" of the combination of the two.

Let it be a fuzzy subset on the universe u, then call it.

The intimacy of the action and. It can be seen that when s0(A, b) is larger (therefore, the larger it is, the smaller it is), the closer it is to.

The closeness describes the closeness between fuzzy sets. In fact, due to the different nature of the problems studied, there are other methods of closeness for further study. However, through the application of several appraisal examples, it is found that the expression of formula (1) is more suitable for real estate appraisal.

2, the principle of proximity

Suppose there are m fuzzy sets on the universe U, which constitute a standard model base. γ (u) is the model to be identified. If i0 exists? {1,2, ..., m}, so that

(2)

It is called closest or merged into a class.

3. The principle of proximity of multiple features

Let there be two families of fuzzy vector sets on the universe u.

Then the proximity of the pair is defined as

(3)

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