Membership truth values

Fuzzy sets and fuzzy logic. Fuzzy sets differ from the normal crisp sets in the fact that their elements have partial membership (represented by a value between 0 an 1) in the set. Fuzzy logic differs from the binary logic by the fact that the truth values are represented by fuzzy sets. 

The input variables state now no longer jumps abruptly from one state to the next, but loses value in one membership function while gaining value in the next. At any one time, the truth value of the indoor or outdoor temperature will almost always be m some degree part of two membership functions ... 

Fuzzy logic is also widely used in process control, because it allows rules to be expressed in a simple linguistic format IF (A) THEN (B), with an associated confidence level that is related to the membership functions. To understand how it is used for control, consider the simple example of a fan heater governed by four rules, summarized in Fig. 8. These rules map onto the four fuzzy sets COLD, COOL, WARM, and HOT also shown in Fig. 8. If the room temperature is 18°C, the heavy line on Fig. 8, then by Rule 3, the fan speed is medium, with truth value 0.7, and by Rule 4, the fan speed is low with truth value 0.3. The process of... 

In fact Baldwin gives special labels to certain truth value fuzzy sets or truth value restrictions such as true, false, unrestricted, impossible, absolutely true and absolutely false. In Fig. 6.12 a set of definitions are illustrated which were those adopted in Baldwin s earlier work. Later these definitions were slightly amended (Fig. 6.20). It is most important to note that we are now dealing with truth value restrictions. Thus the membership of any element of a given truth value restriction will be the maximum possible or least restrictive value, given the available information. 

The truth value of a proposition is calculated by a combination of membership degrees. For... 

P(fp) is considered as the weight of the pth rule (Wrp) where P(fp) is PAE of fp (continuous path in the pth rule). Fire strength or membership degree of the pth rule, is equal to the truth value of the proposition which is ... 

In accordance with the membership degrees of different membership functions, the truth values of different rules () can be determined using the following calculations. 

It can be verified that the relations above reduce to their usual counterparts when applied to binary logic. The standard Lukasiewicz logic Li is isomorphic to fuzzy set theory based on the standard fuzzy operations in the same way the two-valued logic is isomorphic to the crisp set theory. The membership degree A(x) for x e X may be interpreted as the truth value of the proposition x is a member of the set A . The reciprocal is also valid. 

The truth value of a rule is determined from the conjunction (i.e. minimum degree of membership of the rule antecedents) (Zadeh (1973)). Thus the trath-value of the rule is taken to be the smallest degree of tmth of the rule antecedents. This tmth-value is then applied to all consequences of the rule. If any fuzzy output is a consequent of more than one rule, that output is set to the highest (maximum) tmth-value of all the mles that include it as a consequent. The result of the mle evaluation is a set of fiizzy conclusions that reflect the effects of all the mles whose tmth-values are greater than zero. 

This has the effect that the rhembership values in the implication relation I must be truth functionally modified by t before the composition is carried out. This is written I(t). The sloping parallel lines in the upper left hand diagram of Fig. 6.14 will then be altered according to the membership levels of t. For example if T is absolutely true as defined in Fig. 6.12, then these lines become horizontal. (You should satisfy yourself that this is so. Try also plotting the lines for t = very true.)... 

Figure 14.21 depicts the curves for five imprecise classes that assign degrees of truth to the input values. In most cases, these curves are triangular, but they can also follow the Gaussian curve or any other mathematical function. In this example, the x-axis represents the absolute input value and the y-axis stands for the degree of truth to the respective membership curve. Any element can thus belong to more than one imprecise set (membership curve). For example, a yarn fineness... 

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