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 certainity .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 :
 

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# 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)

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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 :
 

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# 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)

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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 :

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# 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)

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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 :

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# 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)

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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}




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