Fuzzy terminologies describes various characteristics about the fuzzy set. This article describes, all the fuzzy terminologies with suitable examples.

Suggested reading: Introduction to fuzzy sets

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

A : Crisp set

A : Fuzzy set

Membership value of element in crisp set is defined by the characteristic function χ (chi), where as the membership value of elements in 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 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 core present in it.

core
Core of fuzzy set

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

Graphically, it is represented as

Boundary of fuzzy set

Normality:

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

In other words, 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 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
Bandwidth of fuzzy set

Fuzzy singleton:

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

|A| = { μA(x) = 1 }

Graphically,

Fuzzy singlton
Fuzzy singleton

Symmetry:

Fuzzy set A is symmetric if its membership function around a center 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 then 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 certain point have membership value 1, and all the elements on right side after certain point have membership value 0.

open left fuzzy set

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

open right fuzzy set

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

closed fuzzy set

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 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

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 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} = 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 !