Fuzzy terminologies: All you need to know

Fuzzy terminologies describe various characteristics of the fuzzy set. This article describes all the fuzzy terminologies with suitable examples.

Suggested reading: Introduction to fuzzy sets

We have discussed some real world fuzzy set scenarios.

Note: As stated earlier, to distinguish the fuzzy set from the crisp set, we will be using a bar under the set letter, i.e.

A : Crisp set

A : Fuzzy set

The membership value of elements in the crisp set is defined by the characteristic function χ (chi), where as the membership value of elements in a fuzzy set is defined by the membership function μ (mu).

For crisp set: χ ∈ { 0, 1 }

For fuzzy set: μ ∈ [0, 1]

Fuzzy terminologies

Fuzzy terminologies define the properties of fuzzy sets. A complete set of fuzzy terminologies is discussed here.

Support:

The support of a fuzzy set A is the set of all points x ∈ X such that μ_A(x) > 0

Support( A ) = { x | μ_A(x) > 0, x ∈ X }

Graphically, we can define support of fuzzy set as,

Note: Support of fuzzy set is its Strong 0-cut (Discussed in the later part of this article)

Core:

The core of a fuzzy set A is the set of all points x ∈ X such that μ_A(x) = 1

Core( A ) = { x | μ_A(x) = 1, x ∈ X }

All fuzzy sets might not have a core present in them.

Height of Fuzzy Set: It is defined as the largest membership value of the elements contained in that set. It may not be 1 always. If the core of the fuzzy set is non-empty, then the height of the fuzzy set is 1.

Boundary:

Boundary comprises those elements x of the universe such that 0 < μ_A(x) < 1

Boundary( A ) = { x | 0 < μ_A(x) < 1 , x ∈ X }

We can treat boundary as the difference between support and core.

Graphically, it is represented as

Normality:

A fuzzy set A is normal if its core is non-empty.

In other words, a fuzzy set is normal if its height is 1

Sub-normal Fuzzy set: For a sub-normal fuzzy set, h( A ) < 1, where h( A ) represents the height of the fuzzy set / highest membership value in the fuzzy set.

Crossover points:

A crossover point of a fuzzy set A is a point x ∈ X at which μ_A(x) = 0.5

Crossover( A ) = { x | μ_A(x) = 0.5 }

Graphically, we can represent it as

Bandwidth:

For a fuzzy set, the bandwidth (or width) is defined as the distance between the two unique crossover points.

Bandwidth( A ) = |x₁ – x₂|

Where, μ_A(x₁) = μ_A(x₂) = 0.5

Graphically,

Fuzzy singleton:

A fuzzy set whose core is a single point in X with μ_A(x) = 1, is called a fuzzy singleton. In other words, if the fuzzy set is having only one element with a membership value of 1, then it is called a fuzzy singleton.

|A| = { μ_A(x) = 1 }

Graphically,

Symmetry:

Fuzzy set A is symmetric if its membership function around a centre point x = c is symmetric

i.e. μ_A(x + c) = μ_A(x – c), ∀x ∈ X

Triangular, Trapezoidal, Gaussian etc. are mostly symmetric. This is more natural to represent the membership than a non-symmetric shape.

Alpha cut:

The α-cut of a fuzzy set A is a crisp set defined by A_α = { x | μ_A(x) ≥ α }

Strong α-cut of a fuzzy set A is a crisp set defined by A_α+ = { x | μ_A(x) > α }

For the above diagram,

• The set A_α=0.2 contains all the elements from x₁ to x_n, including both end values
• The set A_α=0.5 contains all the elements from x₂ to x_m, including both end values
• The set A_α=1.0 contains all the elements from x₃ to x_k, including both end values

For different values of α, we get different crisp sets. In general, if α₁ > α₂ then A_α1 ⊆ A_α2

Cardinality:

Open and Closed fuzzy sets:

Open left: As the name suggests, open left fuzzy sets have all the elements on left after a certain point have a membership value of 1, and all the elements on the right side after a certain point have a membership value of 0.

Open right: Open right fuzzy sets have all the elements on left after a certain point have a membership value of 0, and all the elements on the right side after a certain point have a membership value of 1.

Closed: Closed fuzzy sets have all the elements on the left or right side after a certain point have a membership value of 0.

The following diagram graphically demonstrates all three kinds of fuzzy sets.

Convexity:

Crisp Set A is convex if (λx₁ + (1 – λ) x₂ ) in A, where λ ∈ [0, 1]

Fuzzy Set A is convex if μ_A( λx₁ + (1 – λ) x₂)) ≥ min⁡(μ_A(x₁) , μ_A(x₂)), where x₁, x₂ ∈ X

In other words, for any elements x, y and z in a fuzzy set A, the relation x < y < z implies that: μ_A(y) ≥ min (μ_A(x), μ_A(z)). If this condition holds for all points, the fuzzy set is called a convex fuzzy set.

Convex fuzzy sets are strictly increasing and then strictly decreasing

A is convex if all its α-level sets are convex

Watch on YouTube: Fuzzy terminologies

Apart from these fuzzy terminologies, Linguistic variables and Hedges are also used to represent the real world concepts and their membership strenth.

Example: Fuzzy terminologies

Let A = { (x₁, 0), (x₂, 0.2), (x₃, 0.5), (x₄, 1), (x₅, 1), (x₆, 1), (x₇, 0.5), (x₈, 0.2), (x₉, 0) }

Find support, core, crossover points, alpha cut and strong alpha cut for α = 0.2, boundary, bandwidth, normality, scalar and relative cardinality of the given fuzzy set.

Solution:

From the above-discussed definition,

• Support:  { (x₂, 0.2), (x₃, 0.5), (x₄, 1), (x₅, 1), (x₆, 1), (x₇, 0.5), (x₈, 0.2) }
• Core:   { (x₄, 1), (x₅, 1), (x₆, 1) }
• Crossover Points:   { (x₃, 0.5), (x₇, 0.5) }
• Alpha Cut_0.2:   { (x₂, 0.2), (x₃, 0.5), (x₄, 1), (x₅, 1), (x₆, 1), (x₇, 0.5), (x₈, 0.2) }
• Strong Alpha Cut_0.2⁺:   { (x₃, 0.5), (x₄, 1), (x₅, 1), (x₆, 1), (x₇, 0.5) }
• Boundary:   { (x₂, 0.2), (x₃, 0.5), (x₇, 0.5), (x₈, 0.2) }
• Bandwidth:   | x₇ – x₃ |
• Normality:  True
• Scalar Cardinality: \[ | \bar{A} | = 4.4 \]
• Relative Cardinality: \[ \frac{| \bar{A} |}{n} = \frac{4.4}{9} = 0.489 \]

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Test Your Knowledge:

For the fuzzy set A = { (x₁, 0), (x₂, 0.3), (x₃, 0.4), (x₄, 0.5), (x₅, 0.8), (x₆, 1), (x₇, 1), (x₈, 1), (x₉, 1), (x₁₀, 0.7), (x₁₁ , 0.5) , (x₁₂, 0.1), (x₁₃, 0) }, find following.

• Support
• Core
• Crossover points
• Alpha cut for α = 0.3
• Strong Alpha cut for α = 0.4
• Boundary
• Normality
• Scalar Cardinality
• Relative Cardinality

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