Fuzzy Properties

Description:-

Properties Related to Union

The properties related to union are :  Identity,  Idempotence,  Commutativity  and  Associativity. 

■  Identity:              

   A ∪Φ  = A

input    =  Equality [SMALL ∪ EMPTY ,  SMALL]

output = True

 A ∪ X = X

input    =  Equality [SMALL ∪UnivrsalSpace ,  UnivrsalSpace]

output  = True

■Idempotence :     

    A ∪ A = A

input    =  Equality [SMALL ∪ SMALL ,  SMALL]

output = True

■Commutativity :   

   A ∪ B  = B ∪ A

input   =  Equality [SMALL ∪ MEDIUM,  MEDIUM ∪ SMALL]

output = True

Associativity:  A ∪ (B∪ C) = (A∪ B) ∪ C 

input  = Equality [Small ∪ (Medium ∪ Big) , (Small ∪ Medium) ∪ Big]

output = True

Fuzzy Set  Small , Medium ,  Big  

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, 0.7  },  {10, 0.4 },  {11, 0}, {12, 0}}

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

Big        = FuzzySet [{{1,0}, {2,0}, {3,0}, {4,0}, {5,0}, {6,0.1},  {7,0.2}, {8,0.4}, {9,0.6}, {10,0.8}, {11,1}, {12,1}}]

Calculate Fuzzy relations :-

(1)  Medium ∪ Big = FuzzySet [{1,0},{2,0}, {3,0}, {4,0.2}, {5,0.5}, {6,0.8},{7,1}, {8, 1}, {9, 0.6},  {10, 0.8}, {11, 1}, {12, 1}]

(2)  Small ∪ Medium = 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}] 

(3)  Small ∪ (Medium ∪ Big) = 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, 0.8},{11, 1}, {12, 1}]

 ( 4) (Small ∪ Medium)  ∪ Big] = 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, 0.8}, {11, 1},{12,1}]

Fuzzy set  (3)   and  (4)  proves  Associativity relation.