1. 2004/10/5fuzzy set theory chap03.ppt1 Classical Set Theory

2. 2004/10/5fuzzy set theory chap03.ppt2 Basic concepts Set: a collection of items To Represent sets –List method A={a, b, c} –Rule method C = { x | P(x) } –Family of sets {A i | i  I } –Universal set X and empty set  The set C is composed of elements x Every x has property P i: index I: index set

3. 2004/10/5fuzzy set theory chap03.ppt3 Set Inclusion A  B : x  A implies that x  B A = B : A  B and B  A A  B : A  B and A  B A is a subset of B A and B are equal set A is a proper subset of B

4. 2004/10/5fuzzy set theory chap03.ppt4 Power set All the possible subsets of a given set X is call the power set of X, denoted by P(X) = {A| A  X} |P(X) | = 2 n when |X| = n X={a, b, c} P(X) = { , a, b, c, {a, b}, {b, c}, {a, c}, X}

5. 2004/10/5fuzzy set theory chap03.ppt5 Set Operations Complement Union Intersection difference

6. 2004/10/5fuzzy set theory chap03.ppt6 Basic properties of set operations

7. 2004/10/5fuzzy set theory chap03.ppt7 function A function from a set A to a set B is denoted by f: A  B –Onto –Many to one –One-to-one

8. 2004/10/5fuzzy set theory chap03.ppt8 Characteristic function Let A be any subset of X, the characteristic function of A, denoted by , is defined by Characteristic function of the set of real numbers from 5 to 10

9. 2004/10/5fuzzy set theory chap03.ppt9 Real numbers Total ordering: a  b Real axis: the set of real number  (x-axis) Interval: [a,b], (a,b), (a,b] One-dimensional Euclidean space

10. 2004/10/5fuzzy set theory chap03.ppt10 two-dimensional Euclidean space The Cartesian product of two real number –   –Plane –Cartesian coordinate, x-y axes Cartesian product –A  B = { | a  A and b  B}

11. 2004/10/5fuzzy set theory chap03.ppt11 Convexity A subset of Euclidean space A is convex, if line segment between all pairs of points in the set A are included in the set. A is convex  r+(1- )s  A,  r,s  A,  [0,1]

12. 2004/10/5fuzzy set theory chap03.ppt12 partition Given a nonempty set A, a family of pairwise disjoint subsets of A is called a partition of A, denoted by  (A), iff the union of these subsets yields the set A.  (A) = {A i | i  I,  A i  A}  A i  A j =  for each pair i  j(i,j  I) and  i  I A i =A Blocks refinement 1, 2, 3, 4, 5 4, 5 3, 4, 51, 2 1, 2, 3 4, 51, 2 3 partition refinement