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where μ, λ ∈ [0, 1]. The first constraint restricts the normalized energy measure of fuzziness as it should not exceed μ (maintaining sufficient specificity). Simultaneously, the probability of the statement needs to be meaningful (greater than λ) - this request is captured through the second condition. The region of feasible linguistically quantified statements is marked in Fig. 3.23.


Figure 3.23  (μ, λ) region of feasible probability - possibility quantified statements

These two requirements are in conflict: less specific statements (fuzzier As) lead to higher probabilities, however their semantics is weakened. On the other hand, too specific statements (whose meaning is more substantial) are not sufficiently supported by a visible probabilistic evidence.

Let us now consider now that A is a set. The previous formulation is read as

The range [x0 - ε, + x0 ε] plays a role of a confidence interval and is commonly utilized in statistical analysis. The higher the probability (their specific values are usually set up to 0.95, 0.99, and 0.999; in fact, the tradition is to use their complements that is 0.05, 0.01, and 0.001), the broader the confidence interval. What occurs at higher confidence levels, particularly at the highest probability equal 0.999 is that the associated confidence intervals get too broad and tend to become almost meaningless.

In addition to fuzzy probabilities one can easily envision some other generalizations (Zadeh, 1968) of the standard probabilistic notions such as, for instance

expected (mean) value of A

variance of A

3.19. Conclusions

We have covered a series of key aspects of fuzzy sets. The chapter serves as a concise introduction to fuzzy sets. The exposition of the material naturally gravitates towards primordial issues of knowledge representation as furnished by the technology of fuzzy quantities. Fuzzy sets are conceptual extensions of the generic set theoretic framework and as such are predominantly geared towards various aspects of knowledge representation, especially knowledge granulation.

3.20. Problems

3.1. Show that the minimum and maximum operators can be expressed in the following equivalent form

min(x, y) = 0.5(|x+y| + |x-y|)
max(x, y) = 0.5(|x+y| - |x-y|)

3.2. Consider the sets A = {x ∈ R | -1 ≤ x ≤ 2 } and B = {x ∈ R | 1 ≤ x ≤ 4} which model two intervals in the real line R.

(a)  Find the characteristic function of A and B.
(b)  Determine the union, intersection, and the complement of sets A and B in terms of their characteristic functions.

3.3. Let A be a fuzzy set with the triangular membership function A(x;-1,0,1) in X = [-2, 2]. Perform the five operations above, show them graphically, and compare the results.

3.4. Find all α-cuts of A = 0.1/2+0.3/3+1.0/4+0.3/5 in X = {1, 2, 3, 4, 5}. Show how to reconstruct. A from these α-cuts. What is the form of this reconstruction (viz. the obtained fuzzy set).

3.5. Assume the fuzzy set A = 1.0/1+0.8/2+0.5/3+0.1/4 defined in the universe X = {1, 2, 3, 4, 5}. Find all of its α-cuts. Show how A can be expressed in terms of the family composed of all of its α-cuts.

3.6. Given the fuzzy set A with the following membership function

(a)  Sketch the graph of the function. What is its type?
(b)  How would you attach a linguistic description of a concept conveyed by A?

3.7. Perform the operations of intersection, union and complement in terms of characteristic functions, considering that A = {x ∈ R | 1 ≤ x ≤ 3}, and B = {x ∈ R | 2 ≤ x ≤ 4}. Show the obtained results in a graphical way.

3.8. Consider two fuzzy sets with triangular membership functions A(x;1,2,3) and B(x;2,2,4).

(i)  Find their intersection and union, express them analytically, using the min and max operators.
(ii)  Exercise on the intersection and union by choosing different t- and s-norms.
(iii)  Find the complement of A and B and intersect them with the original sets using several t-norms.

Repeat the same exercise with the union operation exploiting different s-norms.

3.9. Show that the drastic sum and drastic product satisfy the law of excluded middle and the law of contradiction.

3.10. Consider the quadratic form of the entropy functional,

Show that the entropy of A is equal (up to a certain constant factor) to the Euclidean distance between this fuzzy set and its 1/2-cut.

3.11. Calculate the energy measure of fuzziness of the fuzzy set defined as a Gaussian membership function

Discuss several forms of the mapping (e) and compare the obtained results.

3.12. Considering triangular fuzzy numbers, calculate the energy measure of fuzziness for e(u) = up. Plot E(A) as a function of “p” by carrying out respective computations.

3.13. For the image given in Fig. 3.24 (refer also to its tabular format below) compute its entropy and energy measure of fuzziness. Modify a brightness level of each pixel according to the formula (contrast intensification)


Figure 3.24  An example image

Calculate entropy and energy of this image. Obtain a series of images by iteratively intensifying contrast. What happens to their entropy and energy measure ?

3.12. Elaborate on a notion of the frame of cognition in the setting of numerical computing. Consider, for instance, a number of cognitive perspectives based on

(i)  intervals
(ii)  integer numbers
(iii)  real numbers

3.13. Show two relevant examples showing when the reconstruction of a fuzzy relation from its projection is successful and when it totally fails.

3.14. Given a collection of four Gaussian-like membership functions of the frame of cognition, these functions are of the form

Assume that m1 = 1, m2 = 4 and σ = 2. Develop a family of the induced shadowed sets.

Discuss a case of variable σ. How does this parameter influence the position and overlap of the induced shadows?


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