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.

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 the 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 = AB

C = AB = {(x, μAB (x)) | ∀x ∈ X}

μC(x) = μAB (x) = μA(x) ∨ μB(x)

= max( μA(x), μB(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 maximum membership value.

Fuzzy union

Example of Fuzzy Union:

C = AB = {(x, μAB (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) }

μAB (x1) = max( μA(x1), μB(x1) ) = max⁡ { 0.2, 0.8 } = 0.8

μAB (x2) = max( μA(x2), μB(x2) ) = max⁡ { 0.5, 0.6 } = 0.6

μAB (x3) = max( μA(x3), μB(x3) ) = max⁡ { 0.6, 0.4 } = 0.6

μAB (x4) = max( μA(x4), μB(x4) ) = max⁡ { 0.8, 0.2 } = 0.8

μAB (x5) = max( μA(x5), μB(x5) ) = max⁡ { 1.0, 0.1 } = 1.0

So, AB = { (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 the 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 = AB

C = AB = {(x, μAB (x)) | ∀x ∈ X}

μC(x) = μAB (x) = μA(x) ⋀ μB(x)

= min( μA(x), μB(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.

Fuzzy intersection

Example of Fuzzy Intersection:

C = AB = {(x, μAB (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) }

μAB (x1) = min( μA(x1), μB(x1) ) = max⁡ { 0.2, 0.8 } = 0.2

μAB (x2) = min( μA(x2), μB(x2) ) = max⁡ { 0.5, 0.6 } = 0.5

μAB (x3) = min( μA(x3), μB(x3) ) = max⁡ { 0.6, 0.4 } = 0.4

μAB (x4) = min( μA(x4), μB(x4) ) = max⁡ { 0.8, 0.2 } = 0.2

μAB (x5) = min( μA(x5), μB(x5) ) = max⁡ { 1.0, 0.1 } = 0.1

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

Complement:

Fuzzy complement is identical to crisp complement operation. 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, is denoted by AC, is defined as

AC = {(x, μAC (x)) | ∀x ∈ X}

AC (x) = 1 – μA(x)

Fuzzy complement

Example of Fuzzy Complement:

AC (x) = 1 – μA(x)

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

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

AAC = { (x1, 0.8), (x2, 0.5), (x3, 0.6), (x4, 0.8), (x5, 1.0) } ≠ X

AAC = { (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 law of excluded middle.

Watch on YouTube: Fuzzy operations

fuzzy operations on youtube

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 !

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10 Responses

  1. mohammadc says:

    helpful article.

  2. Parekh Payal says:

    Excellent explanation

  3. suhail says:

    Graphically we can represent union operation as follow. this sentence has a mistake please correct it . it should be “intersection ” in the place of “Union”  

  4. Alachew Amaneh says:

    How to show union is associative in fuzzy set

  5. Alachew Amaneh says:

    Please tell me how we prove that union is associtive

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