Fuzzy Set

Introduction:-A Fuzzy Set is any set that allows its members to have different degree  of  membership,  called membership function,  in  the  interval [0 , 1].  

A  fuzzy set  A,  defined in the universal space  X,   is a function defined  in  X  which  assumes values  in  the  range [0, 1]. 

A fuzzy set A is written as a set of pairs  {x, A(x)}  as 

          A = {{x ,  A(x)}} ,   x in the set X 

where   x   is  an  element  of  the universal space X,   and   A(x) is  the value  of  the  function A  for  this  element. 

The value A(x)  is the  membership  grade of the element  x  in a fuzzy set  A.

Example :  Set  SMALL  in set X  consisting  of  natural numbers  ≤  to 12.

 Assume:  SMALL(1) = 1,      SMALL(2) = 1,     SMALL(3) = 0.9,  SMALL(4) = 0.6,

 SMALL(5) = 0.4,  SMALL(6) = 0.3,  SMALL(7) = 0.2,  SMALL(8) = 0.1,

SMALL(u) = 0 for u >= 9. 

Set SMALL = {{1, 1   },   {2, 1  },  {3, 0.9},   {4, 0.6},   {5, 0.4},  {6, 0.3},  {7, 0.2}, {8, 0.1},    {9, 0  },  {10, 0 },   {11, 0},    {12, 0}}

A fuzzy set can be defined precisely by associating with each x ,  its  grade of membership in SMALL.

Universal space for fuzzy sets in fuzzy logic was  defined only on the integers.  Now,  the universal space for fuzzy sets  and fuzzy relations is defined with three numbers.   The first  two numbers specify the start and end of the universal space,  and  the  third argument specifies the increment between elements.  This  gives  the user more flexibility in choosing the universal space.

Example  :   The fuzzy set  of  numbers, defined  in  the  universal space    X = { xi } = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}   is presented as   SetOption [FuzzySet,  UniversalSpace → {1, 12, 1}]