60.

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Figure 4.1 (a) In the traditional crisp set concept, the mambership grade of cluster a1 or a2 is either 0 or 1. (b) In fuzzy set theory, overlap between the two clusters is allowed. See text for details.

(4.3)

Note that, in Equation (4.3), the symbol ‘/’ does not refer to the division operator. It is used to represent the link between the value of s and its corresponding membership grade μ_G(s) in the fuzzy subset G. As the value of μ_G(s) approaches unity, the greater is the chance that s belongs to G. For example, suppose that a very simple universe of discourse S contains only three pixels and the membership function μ_G for fuzzy subset G is defined by:

(4.4)

where denotes the grey value of each pixel. The fuzzy subset G can be represented as:

(4.5)

The term 0.44/5 is interpreted as ‘a pixel with grey value 5 has a membership grade of 0.44 in G’.

The height of a fuzzy subset G denoted by height (G) is defined as the highest membership value contained in that fuzzy subset. For example, Equation (4.5) shows the height of fuzzy subset G is height (G)=1.

The α-cut of a fuzzy subset generates a crisp set in which the universe of discourse has the membership grades equal to or greater than a. Thus, the α-cut of a fuzzy subset G, denoted by G_α, can expressed by:

(4.6)

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