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 :
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) |
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) |
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) |
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}
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