# 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 :

## 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}

```

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