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

### Union:

In case of union of crisp sets, we simply have to select repeated elements only once. In case of fuzzy sets, when there are common elements in both the fuzzy sets, we should select the element with 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 operation as follow. Red and Blue membership functions represents the fuzzy value for elements in set A and B, respectively. Wherever these fuzzy functions overlaps, we have to consider the point with maximum membership value.

### 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 case of intersection of crisp sets, we simply have to select common elements from both the sets. In 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 union operation as follow. Red and Blue membership functions represents the fuzzy value for elements in set A and B, respectively. Wherever these fuzzy functions overlaps, we have to consider the point with minimum membership value.

### 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)

### 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.

## 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) }