This article talks about the fuzzy set example. As discussed in the previous article about fuzzy set, many natural phenomenon can easily be modeled using fuzzy set. Some of the examples are discussed here to make the understanding more clear.

Crisp representation has limitation of not being able to incorporate partial membership of element. In real world, many times, element may belong to multiple sets with different membership values. Specifically, when we represent concepts using linguistic variables.

For example, if we have three linguistic representation for the temperature – called cool, nominal and warm, then depending upon value, temperature ‘t’ has different membership in all three sets. The temperature value t = 5 might have membership value 0.7 in cool set, 0.3 in nominal set and 0 in warm set. Because, 5 degree is considered quite cool but its not warm at all, so its membership value in cool set would be high, where is it will be very low in warm set

Few examples of fuzzy set are discussed here for better understanding.

## Example – 1:

Let us try to represent the concept 2 or so using fuzzy set. We can use different functions to model this concept. Same number can take different membership values (fuzzy value) based on the membership function used to assign the membership to the number.

Following figure represents the concept 2 or so using three different membership function. All three functions are modelled using different gaussians. We can observe the membership value of elements 1, 2, 3 and 4 for each of the membership function

A = { 2 or so }

Membership value of few elements for fuzzy set A is,

μA(1) = 0.5, μA(2) = 1.0, μA(3) = 0.5, μA(4) = 0.0

In set notation, we can combine them all together as,

\bar{A} = \{ \frac{0.5}{1} + \frac{1.0}{2} + \frac{0.5}{3} + \frac{0.0}{4} \}

In this representation, denominator denotes element of the sets, and numerator denotes membership value of corresponding element.

Alternatively, we can represent the fuzzy set using tuple notation as,

A = { (1, 0.3), (2,1.0), (3, 0.3), (4, 0.0) }

In each tuple, first value represents element of the set, and second value represents its membership in the set.

In similar way, the membership value assigned by second function would be,

B = { (1, 0.7), (2,1.0), (3, 0.7), (4, 0.2) }

And finally, the third function will assign membership value to the elements as shown below:

C = { (1, 0.9), (2,1.0), (3, 0.9), (4, 0.4) }

As we can see that, same elements have different membership value for different function.

It is quite obvious from the shape of the gaussians that if element is in the center of gaussian, it will be assigned highest membership value, and as we move away, the membership value decreases. As the spread of gaussian increases, the more and more elements fall in the set and the elements near to edges assigned smaller and smaller membership value.

## Example – 2:

Let us consider age is to be represented using fuzzy set. We will be using two fuzzy sets Young and Very Young to represent different age range .

A = {young} ∈ [0, 90]

B = {Very Young} ∈ [0, 60]

Let us see, how the membership value is affected by two different functions. As of now, the mathematical description of both fuzzy sets is not important, but as a human, we can easily map that if the range of young set is from 0 to 90, then as we move away from 0, they youngness will keep decreasing and become 0 at the age 90. The age 30 is in upper half of the entire range and hence it has higher membership value for being considered as young.

For set Very Young, the range is from 0 to 60, so age 30 is at the center and hence its obvious it takes membership value 0.5.

As we can see from the diagram, person with age 30 has membership 0.9 in set Young, where as it is 0.5 in set Very Young.

## Example – 3:

Let us represent the concept real number close to 0 by fuzzy set A. This concept can be modeled using following function.

This is symmetric function, and it is modelled such that for x = 0, it returns the highest membership value, i.e. 1, and as we move away on either side, it reduces the membership value with equally in both directions. So the membership value of +x and -x would be same.

As number moves away from 0 in either direction, its membership value decreases gradually. And after some range, it vanishes to 0. Rest of all elements are far from 0, and hence we can assign them 0 membership value in this particular set.

## Example – 4:

Let us modify the concept discussed in previous example. Let fuzzy set A represents the Real number very close to zero. It is not hard to understand now that the membership value of element will decrease sharply compared to previous example, as the value of element go away from the zero.

To model the fuzzy set real number very very close to zero, we can take cube of the numerator term, which will further reduces the span of gaussian function. Thus, fuzzy is very natural way to represent the things where we want to incorporate partial membership with varying degree.

## Test Your Knowledge:

How would you represent following concept?

1. A waste is only toxic if it has an oral toxicity greater than 500“. Show the crisp and fuzzy set representation when the waste is orally toxic
2. Show fuzzy and crisp representation for set of “good students” having a CGPA 5.0 and above