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)