Fuzzy Set Theory Background

Fuzzy Set Theory (FST) was formalised by Prof. Lofti Zadeh at the University of California in 1965. The significance of fuzzy variables is that they facilitate gradual transition between states and consequently, possess a natural capability to express and deal with observation and measurement uncertainties. 

The limiting feature of bivalent sets is that they are mutually exclusive - it is not possible to have a membership of more than one set. It is not accurate to define a transition from a quantity such as warm to hot . In the real world a smooth (unnoticeable) drift fiom warm to hof would occur. The natural phenomenon can be described more accurately by FST. Rgure 6.3 shows how the same information can be quantified using fuzzy sets to describe this natural drift 

A set A, with points or objects in some relevant universe, X, is defined as these elements of x that satisfy the membership property defined for A. hi traditional crisp sets theory each element of X either is or is not an element of A. Elements in a fuzzy set (denoted by, eg A) can have a continuum of degrees of membership ranging from complete membership to complete nonmembership (Zadeh (1987)). 

The use of a numerical scale for the degree of membership provides a convenient way to represent gradation in the degree of membraship. Precise degrees of membership generally do not exist. Instead they tend to reflect sometimes subjective ordering of the element in the universe. 

Fuzzy sets can be represented by various shapes. They are commonly represented by S-curves, ti-curves, triangular curves and linear curves. The shape of the fiizzy set depends on the best way to represent the data. In general the membership (often indicated on the vertical axis) starts at 0 (no membership) and continues to 1 (full membership). The domain of a set is indicated along the horizontal axis. The fuzzy set shape defines the relationship between the domain and the membership values of a set. 

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