Discrete membership functions

Discrete Membership Functions Often specification of continuous membership functions does not make sense. If, for example, the qualification of a sample state for a spectrophotometric analysis is to be described, discrete values, such as solid, gas, or liquid, are required. 

Discrete membership functions in relation to the fuzzy set A are represented as pairs consisting of the variable value and the... 

Suppose we have linguistic variables temperature (f) and yield (y). The discrete membership functions are given in Table 1, rows /Z/i(f), Ms(> )- The conditional fuzzy proposition... 

For an ordinary subset Y of a set X,X D Y, an element. x of X is either a member or not a member of the subset Y hence, by definition, the membership function urix) of elements x of X can take only two discrete values, 1 or 0. Such ordinary sets are often referred to as crisp sets . 

The aim of the a-level optimization is the computation of the membership function of the fuzzy result value f. Therefore, the min-max operator of the extension principle (Eq. 26) is replaced by optimization tasks. The necessary discretization of the fuzzy input quantities into a-level is defined in Eq. 5. The main idea of this approach is that the necessary information is the minimum and maximum value for the response only, which can be reached at each a-level a.k G (0,1]. The membership values of the result are... 

The concept of fuzzy sets extends this crisp assignment by allowing membership values between 0 and 1. These fuzzy sets are normalized to the interval [0,1]. They can be represented either as discrete or by means of a function (cf. Eqs. 8.36-8.39). 

The necessity of such a convention—and ultimately the underiying cause of statelessness—lay in the fact that at the core of sovereignty was the unfettered discretion to set the terms of membership. In other words, states have traditionally been free to establish their own rules for the acquisition and loss of nationality, in accordance with their own interests and ideology. Where such rules are exclusionary or there is a corrflict between the rales of different states, statelessness may result The function then, of what would become the 1961 Convention on the Reduction of Statelessness, was to agree some restrictions on states freedom in rrationaHty matters. 

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