Fuzzy operations – Explained with examples

Fuzzy operations are performed on fuzzy sets, whereas crisp operations are performed on crisp sets. Fuzzy operations are very useful in the design of a Fuzzy Logic Controller. It allows the manipulation of fuzzy variables by different means.

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Suugested Reading: Fuzzy Terminologies

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Union:

In the case of the union of crisp sets, we simply have to select repeated elements only once. In the case of fuzzy sets, when there are common elements in both fuzzy sets, we should select the element with the maximum membership value.

The union of two fuzzy sets A and B is a fuzzy set C, written as C = A ∪ B

C = A ∪ B = {(x, μ_{A ∪ B} (x)) | ∀x ∈ X}

μ_C(x) = μ_{A ∪ B} (x) = μ_A(x) ∨ μ_B(x)

= max( μ_A(x), μ_B(x) ), ∀x ∈ X

Graphically, we can represent union operations as follows: Red and Blue membership functions represent the fuzzy value for elements in sets A and B, respectively. Wherever these fuzzy functions overlap, we have to consider the point with the maximum membership value.

Example of Fuzzy Union:

C = A ∪ B = {(x, μ_{A ∪ B} (x)) | ∀x ∈ X}

A = { (x₁, 0.2), (x₂, 0.5), (x₃, 0.6), (x₄, 0.8), (x₅, 1.0) }

B = { (x₁, 0.8), (x₂, 0.6), (x₃, 0.4), (x₄, 0.2), (x₅, 0.1) }

μ_{A ∪ B} (x₁) = max( μ_A(x₁), μ_B(x₁) ) = max⁡ { 0.2, 0.8 } = 0.8

μ_{A ∪ B} (x₂) = max( μ_A(x₂), μ_B(x₂) ) = max⁡ { 0.5, 0.6 } = 0.6

μ_{A ∪ B} (x₃) = max( μ_A(x₃), μ_B(x₃) ) = max⁡ { 0.6, 0.4 } = 0.6

μ_{A ∪ B} (x₄) = max( μ_A(x₄), μ_B(x₄) ) = max⁡ { 0.8, 0.2 } = 0.8

μ_{A ∪ B} (x₅) = max( μ_A(x₅), μ_B(x₅) ) = max⁡ { 1.0, 0.1 } = 1.0

So, A ∪ B = { (x₁, 0.8), (x₂, 0.6), (x₃, 0.6), (x₄, 0.8), (x₅, 1.0) }

Intersection:

In the case of the intersection of crisp sets, we simply have to select common elements from both sets. In the case of fuzzy sets, when there are common elements in both fuzzy sets, we should select the element with minimum membership value.

The intersection of two fuzzy sets A and B is a fuzzy set C, written as C = A ∩ B

C = A ∩ B = {(x, μ_{A ∩ B} (x)) | ∀x ∈ X}

μ_C(x) = μ_{A ∩ B} (x) = μ_A(x) ⋀ μ_B(x)

= min( μ_A(x), μ_B(x) ), ∀x ∈ X

Graphically, we can represent the intersection operation as follows: Red and blue membership functions represent the fuzzy value for elements in sets A and B, respectively. Wherever these fuzzy functions overlap, we have to consider the point with the minimum membership value.

Example of Fuzzy Intersection:

C = A ∩ B = {(x, μ_{A ∩ B} (x)) | ∀x ∈ X}

A = { (x₁, 0.2), (x₂, 0.5), (x₃, 0.6), (x₄, 0.8), (x₅, 1.0) }

B = { (x₁, 0.8), (x₂, 0.6), (x₃, 0.4), (x₄, 0.2), (x₅, 0.1) }

μ_{A ∩ B} (x₁) = min( μ_A(x₁), μ_B(x₁) ) = max⁡ { 0.2, 0.8 } = 0.2

μ_{A ∩ B} (x₂) = min( μ_A(x₂), μ_B(x₂) ) = max⁡ { 0.5, 0.6 } = 0.5

μ_{A ∩ B} (x₃) = min( μ_A(x₃), μ_B(x₃) ) = max⁡ { 0.6, 0.4 } = 0.4

μ_{A ∩ B} (x₄) = min( μ_A(x₄), μ_B(x₄) ) = max⁡ { 0.8, 0.2 } = 0.2

μ_{A ∩ B} (x₅) = min( μ_A(x₅), μ_B(x₅) ) = max⁡ { 1.0, 0.1 } = 0.1

So, A ∩ B = { (x₁, 0.2), (x₂, 0.5), (x₃, 0.4), (x₄, 0.2), (x₅, 0.1) }

Complement:

Fuzzy complement is identical to crisp complement operation. The membership value of every element in the fuzzy set is complemented with respect to 1, i.e. it is subtracted from 1.

The complement of fuzzy set A, denoted by A^C, is defined as

A^C = {(x, μ_A^C (x)) | ∀x ∈ X}

A^C (x) = 1 – μ_A(x)

Example of Fuzzy Complement:

A^C (x) = 1 – μ_A(x)

A = { (x₁, 0.2), (x₂, 0.5), (x₃, 0.6), (x₄, 0.8), (x₅, 1.0) }

A^C = { (x₁, 0.8), (x₂, 0.5), (x₃, 0.4), (x₄, 0.2), (x₅, 0.0) }

A ⋃ A^C = { (x₁, 0.8), (x₂, 0.5), (x₃, 0.6), (x₄, 0.8), (x₅, 1.0) } ≠ X

A ∩ A^C = { (x₁, 0.2), (x₂, 0.5), (x₃, 0.4), (x₄, 0.2), (x₅, 0.0) } ≠ Φ

Unlike crisp sets, fuzzy sets do not hold the law of contradiction and the law of excluded middle.

Watch on YouTube: Fuzzy operations

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Test Your Knowledge:

For following fuzzy sets, perform union, complement and intersection operations.

A = { (x₁, 0.4), (x₂, 0.5), (x₃, 0.2), (x₄, 0.4), (x₅, 0.8) }

B = { (x₁, 1.0), (x₂, 0.3), (x₃, 0.5), (x₄, 0.7), (x₅, 0.1) }

Please post your answer / query / feedback in comment section below !