# Fuzzy terminologies: All you need to know

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 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 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 core present in it.

**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 < μ_{A}(x) < 1

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

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

Graphically, it is represented as

### 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.

### 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 ) = |x_{1} – x_{2}|

Where, μ_{A}(x_{1}) = μ_{A}(x_{2}) = 0.5

Graphically,

### Fuzzy singleton:

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

Graphically,

### Symmetry:

Fuzzy set A is symmetric if its membership function around a center 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 then 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 x_{1}to x_{n}, including both end values - The set A
_{α=0.5}contains all the elements from x_{2}to x_{m}, including both end values - The set A
_{α=1.0}contains all the elements from x_{3}to x_{k}, including both end values

For different values of α , we get different crisp sets. In general, if α_{1} > α_{2} then A_{α1} ⊆ A_{α2}

### 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 | = Σ_{x ∊ X} { μ_{A}(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 , μ_{A}_{α}(x)) }

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

### 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 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.

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

Following diagram graphically demonstrates all three kinds of fuzzy sets.

### Convexity:

Crisp Set A is convex if (λx_{1} + (1 – λ) x_{2} ) in A, where λ ∈ [0, 1]

Fuzzy Set A is convex if μ_{A}( λx_{1} + (1 – λ) x_{2})) ≥ min(μ_{A}(x_{1}) , μ_{A}(x_{2})), where x_{1}, x_{2} ∈ 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 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

## Example: Fuzzy terminologies

Let A = { (x_{1}, 0), (x_{2}, 0.2), (x_{3}, 0.5), (x_{4}, 1), (x_{5}, 1), (x_{6}, 1), (x_{7}, 0.5), (x_{8}, 0.2), (x_{9}, 0) }

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: { (x
_{2}, 0.2), (x_{3}, 0.5), (x_{4}, 1), (x_{5}, 1), (x_{6}, 1), (x_{7}, 0.5), (x_{8}, 0.2) } - Core: { (x
_{4}, 1), (x_{5}, 1), (x_{6}, 1) } - Crossover Points: { (x
_{3}, 0.5), (x_{7}, 0.5) } - Alpha Cut
_{0.2}: { (x_{2}, 0.2), (x_{3}, 0.5), (x_{4}, 1), (x_{5}, 1), (x_{6}, 1), (x_{7}, 0.5), (x_{8}, 0.2) } - Strong Alpha Cut
_{0.2}^{+}: { (x_{3}, 0.5), (x_{4}, 1), (x_{5}, 1), (x_{6}, 1), (x_{7}, 0.5) } - Boundary: { (x
_{2}, 0.2), (x_{3}, 0.5), (x_{7}, 0.5), (x_{8}, 0.2) } - Bandwidth: | x
_{7}– x_{3}| - 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 = { (x_{1}, 0), (x_{2}, 0.3), (x_{3}, 0.4), (x_{4}, 0.5), (x_{5}, 0.8), (x_{6}, 1), (x_{7}, 1), (x_{8}, 1), (x_{9}, 1), (x_{10}, 0.7), (x_{11} , 0.5) , (x_{12}, 0.1), (x_{13}, 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 !**

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

Thank you.. Thanks for the appreciation

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

Right. Thumbs up

True

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

Thats perfectly right Ebraheem.. Good work

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

Thank you very much. It means a lot