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 μA(x) > 0

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

Graphically, we can define support of fuzzy set as,

support
Support of fuzzy set

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 μA(x) = 1

Core( A ) = { x | μA(x) = 1, x ∈ X }

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

core
The core of fuzzy set

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 < μA(x) < 1

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

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

Graphically, it is represented as

The boundary of fuzzy set

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.

normality of fuzzy set
Normality of fuzzy set

Crossover points:

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

Crossover( A ) = { x | μA(x) = 0.5 }

Graphically, we can represent it as

Cross-over points of fuzzy set

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, μA(x1) = μA(x2) = 0.5

Graphically,

bandwidth of fuzzy set
The bandwidth of fuzzy set

Fuzzy singleton:

A fuzzy set whose core is a single point in X with μA(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| = { μA(x) = 1 }

Graphically,

Fuzzy singlton
Fuzzy singleton

Symmetry:

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

i.e. μA(x + c) = μA(x – c), ∀x ∈ X

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

Symmetric fuzzy sets
Symmetric fuzzy sets

Alpha cut:

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

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

Alpha cut in fuzzy set
Alpha cut in fuzzy set

For the above diagram,

  • The set Aα=0.2 contains all the elements from x1 to xn, including both end values
  • The set Aα=0.5 contains all the elements from x2 to xm, including both end values
  • The set Aα=1.0 contains all the elements from x3 to xk, including both end values

For different values of α, we get different crisp sets. In general, if α1 > α2 then Aα1 ⊆ Aα2

effect of alpha on alpha cut sets
Relation between different α values

Cardinality:

Scalar cardinality:

Scalar cardinality is defined by the summation of membership values of all elements in the set. For the data given in the table,

| A | = Σx ∊ X { μA(x) }

|Senior| = 0.3 + 0.9 + 1 + 1 = 3.2

Relative cardinality:

|| A || = | A | / | X |

|| Senior || = 3.2 / 9 = 0.356

Fuzzy cardinality:

| A |F = { (α , μAα(x)) }

| Senior |F = { (4, 0.3), (3, 0.9), (2, 1.0) }

cardinality example
cardinality example

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 left fuzzy set

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.

open right fuzzy set

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

closed fuzzy set

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

Open and closed fuzzy sets
Open and closed 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 μA( λx1 + (1 – λ) x2)) ≥ min⁡(μA(x1) , μA(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: μA(y) ≥ min (μA(x), μA(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

Convex fuzzy set (left), Non-convex fuzzy set (right)

Watch on YouTube: Fuzzy terminologies

fuzzy terminologies - on youtube

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) }

fuzzy terminologies example
Fuzzy Terminologies – all together

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 Cut0.2:   { (x2, 0.2), (x3, 0.5), (x4, 1), (x5, 1), (x6, 1), (x7, 0.5), (x8, 0.2) }
  • Strong Alpha Cut0.2+:   { (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: \[ \frac{| \bar{A} |}{n} = \frac{4.4}{9} = 0.489 \]

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 !

11 Responses

  1. Rajvi Upadhyay says:

    Very to the point and resourceful content. The video makes it more easier to understand. Great work.

  2. Smita Mahajani says:

    Core of A={(x6,1), (x7,1) ,(x8,1) , (x9,1)}

  3. Ebraheem says:

    Support A = { (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)}
    Core A = { (x6, 1), (x7, 1), (x8, 1), (x9, 1), (x12, 0.1)}
    Crossover A = { (x4, 0.5), (x11 , 0.5) }
    Alpha cut for α = 0.3 = {(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)}
    Strong Alpha cut for α = 0.4 = { (x4, 0.5), (x5, 0.8), (x6, 1), (x7, 1), (x8, 1), (x9, 1), (x10, 0.7), (x11 , 0.5)}
    Boundary A = { (x2, 0.3), (x3, 0.4), (x4, 0.5), (x5, 0.8), (x10, 0.7), (x11 , 0.5) , (x12, 0.1)}
    Normality A = True 
    Scalar Cardinality A = 7.3
    Relative Cardinality A = 7.3 / 13= 0.561

  4. Rajpreet Singh says:

    The articles are really helpful. Thank You for this amazing content.

  5. Manas says:

    nyc explanation

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