SKEDSOFT

Example 2.

The empty set  Φ    and  the  universal set  X,  as fuzzy sets, are  complements of one another.    

                     Φ' = X        ,       X' = Φ

The fuzzy set  B  EMPTY

Empty =  FuzzySet {{1, 0   },  {2, 0  },  {3, 0},   {4, 0},   {5, 0},  {6, 0}, {7, 0},     {8, 0},    {9, 0  },  {10, 0 },   {11, 0},    {12, 0}}

The fuzzy set  A  UNIVERSAL

Universal =  FuzzySet {{1, 1   },   {2, 1  },  {3, 1},   {4, 1},   {5, 1},  {6, 1}, {7, 1},  {8, 1},    {9, 1  },  {10, 1 },   {11, 1},    {12, 1}}

The fuzzy operation :  Compliment  

EMPTY  =  Compliment [UNIVERSALSPACE]

Union :-Let A and B  be fuzzy sets defined in the space X.  The union is defined as the smallest fuzzy set that contains both A and B.  The union of A and B is denoted by A ∪B. 

The following relation must be satisfied for the union operation :

for all x in the set X,    (A ∪B)(x) = Max (A(x), B(x)).

Fuzzy Union :   (A ∪B)(x)  =  max [A(x), B(x)]      for   all   x  ∈X

Example 1 :  Union of  Fuzzy  A  and  B

     A(x)  = 0.6    and    B(x)  = 0.4        ∴   (A ∪B)(x)  =  max [0.6, 0.4]   = 0.6

Example 2 :Union of  SMALL and MEDIUM

The fuzzy set  A  SMALL

SMALL =  FuzzySet {{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}}

The fuzzy set  B  MEDIUM

MEDIUM =  FuzzySet {{1, 0   },   {2, 0  },  {3, 0},   {4, 0.2},   {5, 0.5},  {6, 0.8}, {7, 1},     {8, 1}, {9, 0.7  },  {10, 0.4 },  {11, 0.1},   {12, 0}}

The fuzzy operation :   Union    

FUZZYUNION = [SMALL  ∪  MEDIUM]   

SetSmallUNIONMedium = FuzzySet [{{1,1},{2,1}, {3,0.9}, {4,0.6}, {5,0.5}, {6,0.8}, {7,1}, {8, 1}, {9, 0.7},  {10, 0.4}, {11, 0.1}, {12, 0}} , UniversalSpace → {1, 12, 1}]

The notion of the union is closely related to that of the connective "or".  Let  A is a class of "Young" men, B is a class of "Bald" men.  If  "David is Young" or "David is Bald,"  then David is associated with the union of A  and B.   Implies David is a member of A∪B.