Common Operations on Fuzzy Set with Example and Code

What is Fuzzy Set ?

Fuzzy refers to something that is unclear or vague . Hence, Fuzzy Set is a Set where every key is associated with value, which is between 0 to 1 based on the certainty .This value is often called as degree of membership. Fuzzy Set is denoted with a Tilde Sign on top of the normal Set notation.

Operations on Fuzzy Set with Code :

1. Union :

Consider 2 Fuzzy Sets denoted by A and  B, then let’s consider Y be the Union of them, then for every member of  A and  B, Y will be:

 degree_of_membership(Y)= max(degree_of_membership(A), degree_of_membership(B)) 

EXAMPLE :

Python3

# Example to Demonstrate the
# Union of Two Fuzzy Sets
A = dict()
B = dict()
Y = dict()

A = {"a": 0.2, "b": 0.3, "c": 0.6, "d": 0.6}
B = {"a": 0.9, "b": 0.9, "c": 0.4, "d": 0.5}

print('The First Fuzzy Set is :', A)
print('The Second Fuzzy Set is :', B)


for A_key, B_key in zip(A, B):
    A_value = A[A_key]
    B_value = B[B_key]

    if A_value > B_value:
        Y[A_key] = A_value
    else:
        Y[B_key] = B_value

print('Fuzzy Set Union is :', Y)

Output

The First Fuzzy Set is : {'a': 0.2, 'b': 0.3, 'c': 0.6, 'd': 0.6}
The Second Fuzzy Set is : {'a': 0.9, 'b': 0.9, 'c': 0.4, 'd': 0.5}
Fuzzy Set Union is : {'a': 0.9, 'b': 0.9, 'c': 0.6, 'd': 0.6}

2. Intersection :

Consider 2 Fuzzy Sets denoted by A and  B, then let’s consider Y be the Intersection of them, then for every member of  A and  B, Y will be:

degree_of_membership(Y)= min(degree_of_membership(A), degree_of_membership(B)) 

EXAMPLE :

Python3

# Example to Demonstrate
# Intersection of Two Fuzzy Sets
A = dict()
B = dict()
Y = dict()

A = {"a": 0.2, "b": 0.3, "c": 0.6, "d": 0.6}
B = {"a": 0.9, "b": 0.9, "c": 0.4, "d": 0.5}

print('The First Fuzzy Set is :', A)
print('The Second Fuzzy Set is :', B)


for A_key, B_key in zip(A, B):
    A_value = A[A_key]
    B_value = B[B_key]

    if A_value < B_value:
        Y[A_key] = A_value
    else:
        Y[B_key] = B_value
print('Fuzzy Set Intersection is :', Y)

Output

The First Fuzzy Set is : {'a': 0.2, 'b': 0.3, 'c': 0.6, 'd': 0.6}
The Second Fuzzy Set is : {'a': 0.9, 'b': 0.9, 'c': 0.4, 'd': 0.5}
Fuzzy Set Intersection is : {'a': 0.2, 'b': 0.3, 'c': 0.4, 'd': 0.5}

3. Complement :

Consider a Fuzzy Sets denoted by A  , then let’s consider Y be the Complement of it, then for every member of  A  , Y will be:

degree_of_membership(Y)= 1 - degree_of_membership(A)

EXAMPLE :

Python3

# Example to Demonstrate the
# Difference Between Two Fuzzy Sets
A = dict()
Y = dict()

A = {"a": 0.2, "b": 0.3, "c": 0.6, "d": 0.6}

print('The Fuzzy Set is :', A)


for A_key in A:
   Y[A_key]= 1-A[A_key]

print('Fuzzy Set Complement is :', Y)

Output

The Fuzzy Set is : {'a': 0.2, 'b': 0.3, 'c': 0.6, 'd': 0.6}
Fuzzy Set Complement is : {'a': 0.8, 'b': 0.7, 'c': 0.4, 'd': 0.4}

4. Difference : 
Consider 2 Fuzzy Sets denoted by A and  B, then let’s consider Y be the Intersection of them, then for every member of  A and  B, Y will be:

degree_of_membership(Y)= min(degree_of_membership(A), 1- degree_of_membership(B)) 

EXAMPLE :

Python3

# Example to Demonstrate the
# Difference Between Two Fuzzy Sets
A = dict()
B = dict()
Y = dict()

A = {"a": 0.2, "b": 0.3, "c": 0.6, "d": 0.6}
B = {"a": 0.9, "b": 0.9, "c": 0.4, "d": 0.5}

print('The First Fuzzy Set is :', A)
print('The Second Fuzzy Set is :', B)


for A_key, B_key in zip(A, B):
    A_value = A[A_key]
    B_value = B[B_key]
    B_value = 1 - B_value

    if A_value < B_value:
        Y[A_key] = A_value
    else:
        Y[B_key] = B_value

print('Fuzzy Set Difference is :', Y)

Output 

The First Fuzzy Set is : {"a": 0.2, "b": 0.3, "c": 0.6, "d": 0.6}
The Second Fuzzy Set is : {"a": 0.9, "b": 0.9, "c": 0.4, "d": 0.5}
Fuzzy Set Difference is : {"a": 0.1, "b": 0.1, "c": 0.6, "d": 0.5}

Class Fuzzy Sets

Python3

class FzSets:

  def __init__(self):
    self.A = dict()
    self.B = dict()

    self.complement_A = dict()
    self.complement_B = dict()
    self.union_AB = dict()
    self.intersection_AB = dict()
    self.differenceAB = dict()
    self.differenceBA = dict()

    self.change_union = False
    self.change_intersection = False
    self.change_complement = False

  def __init__(self,A,nA,B,nB):
    self.A = A
    self.B = B
    self.Aname = nA
    self.Bname = nB

    self.complement_A = dict()
    self.complement_B = dict()
    self.union_AB = dict()
    self.intersection_AB = dict()
    self.differenceAB = dict()
    self.differenceBA = dict()

    self.change_union = False
    self.change_intersection = False
    self.change_complement = False

  def unionOp(self):
    if self.change_union:
      print('Result of UNION operation :',self.union_AB)
    else:

      #unionSet = set(self.A.keys()).union(self.B.keys())
      sa = set(self.A.keys())
      sb = set(self.B.keys())
      intersectionSet = set(self.A.keys()).intersection(self.B.keys())

      for i in intersectionSet:
        self.union_AB[i] = max(self.A[i],self.B[i])
      for i in sa-intersectionSet:
        self.union_AB[i] = self.A[i]
      for i in sb-intersectionSet:
        self.union_AB[i] = self.B[i]

      print('Result of UNION operation :',self.union_AB)


  def intersectionOp(self):
    if self.change_intersection:
      print('Result of INTERSECTION operation :\n\t\t',self.intersection_AB)
    else:

      #unionSet = set(self.A.keys()).union(self.B.keys())
      sa = set(self.A.keys())
      sb = set(self.B.keys())
      intersectionSet = set(self.A.keys()).intersection(self.B.keys())

      for i in intersectionSet:
        self.intersection_AB[i] = min(self.A[i],self.B[i])
      for i in sa-intersectionSet:
        self.intersection_AB[i] = 0.0
      for i in sb-intersectionSet:
        self.intersection_AB[i] = 0.0

      print('Result of INTERSECTION operation :\n\t\t',self.intersection_AB)
      self.change_intersection = True

  def complementOp(self):
    if self.change_complement:
      print('Result of COMPLEMENT on ',self.Aname,' operation :',self.complement_A)
      print('Result of COMPLEMENT on ',self.Bname,' operation :',self.complement_B)
    else:

      for i in self.A:
        self.complement_A[i] = 1 - A[i]
      for i in self.B:
        self.complement_B[i] = 1 - B[i]

      print('Result of COMPLEMENT on ',self.Aname,' operation :',self.complement_A)
      print('Result of COMPLEMENT on ',self.Aname,' operation :',self.complement_B)

      self.change_complement = True

  def __oneMinustwo(self,L,R):
    minus_d = dict()
    Rcomp = dict()
    for i in R:
      Rcomp[i] = 1 - R[i]
    sa = set(L.keys())
    sb = set(R.keys())
    intersectionSet = sa.intersection(sb)   # min( A , complement(B) )

    # l - r OR a - b
    for i in intersectionSet:
      minus_d[i] = min(L[i],Rcomp[i])
    for i in sa-intersectionSet:
      minus_d[i] = 0.0
    for i in sb-intersectionSet:
      minus_d[i] = 0.0

    return minus_d

  def AminusB(self):
    self.differenceAB = self.__oneMinustwo(self.A,self.B)
    print('Result of DIFFERENCE ',self.Aname,' | ',self.Bname,' operation :\n\t\t',self.differenceAB)

  def BminusA(self):
    self.differenceBA = self.__oneMinustwo(self.B,self.A)
    print('Result of DIFFERENCE ',self.Bname,' | ',self.Aname,' operation :\n\t\t',self.differenceBA)

  def change_Setz(self,A,B):
    self.A = A
    self.B = B

    print('\nSet ',self.Aname,' :',self.A)
    print('Set ',self.Bname,' :',self.B,end='')

    self.change_union = True
    self.change_intersection = True
    self.change_complement = True
    print('\t\t\t Cache Reset')

  def displaySets(self):
    print('\nSet ',self.Aname,' :',self.A)
    print('Set ',self.Bname,' :'  ,self.B)