# 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 ) = |x_{1} – x_{2}|

Where, μ_{A}(x_{1}) = μ_{A}(x_{2}) = 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 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 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 (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 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 = { (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 the 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: \[ \frac{| \bar{A} |}{n} = \frac{4.4}{9} = 0.489 \]

## 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

nyc explanation

Thanks Manas

Dear Sir,

Thankyou for this great knowledge sharing.

Im a bit confuse.

In the part of cardinality, fuzzy cardinality.

I can see |Aa| 4,3,2 in the table.

How to decide this value ? and the value represent for what ?