FUZZY SETS-Characteristics of Crisp Set
المؤلف:
Kwang H. Lee
المصدر:
First Course on Fuzzy Theory and Applications
الجزء والصفحة:
5-7
4-7-2016
1122
Characteristics of Crisp Set
1.1 Ordinary Characteristics
Let us look over the operational characteristics of union, intersection, and complement set [Table 1.1]. Commutativity of union and intersection is
satisfied as follows
The operations of intersection and union follows the associativity
Union or intersection between itselves is reduced to the set itself. This is ‘idempotency’.
In addition, for union and intersection, the distributivity is held.
De Morgan’s law is satisfied with the union, intersection and complement operation.
Table 1.1. Features of Crisp Set
Table 1.1. (cont’)
1.2 Convex Set
Definition (Convex set) The term convex is applicable to a set A in Rn(n-dimensional Euclidian vector space) if the followings are satisfied.
i) Two arbitrary points s and r are defined in A.
(N is a set of positive integers)
ii) For arbitrary real number λ between 0 and 1, point t is involved in A where t is
Fig. 1.1. Convex sets A1, A2, A3 and non-convex sets A4, A5, A6 in |R2
In other wads, if every point on the line connecting two points s and r in A is also in A. (Fig 1.1) shows some examples of convex and non-convex
Sets
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Kwang H. Lee, First Course on Fuzzy Theory and Applications, 2005, Springer, pag(5-7)
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