Fuzzy set example – Real world problem representation

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

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

For example, if we have three linguistic representations 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 a membership value of 0.7 in the cool set, 0.3 in the nominal set and 0 in the warm set. Because, 5 degrees is considered quite cool but it’s not warm at all, its membership value in the cool set would be high, whereas it will be very low in the warm set

A few examples of fuzzy sets are discussed here for better understanding.

Example – 1:

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

The following figure represents concept 2 or so using three different membership functions. 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 }

The membership value of a few elements for fuzzy set A is,

μ_A(1) = 0.5, μ_A(2) = 1.0, μ_A(3) = 0.5, μ_A(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, the denominator denotes an element of the sets, and the numerator denotes the membership value of the 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, the first value represents the element of the set, and the second value represents its membership in the set.

In a similar way, the membership value assigned by the 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, the same elements have different membership values for different functions.

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

Example – 2:

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

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 the young set is from 0 to 90, then as we move away from 0, the youngness will keep decreasing and become 0 at the age 90. The age of 30 is in the upper half of the entire range and hence it has a higher membership value for being considered young.

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

As we can see from the diagram, a person with the age of 30 has membership 0.9 in set Young, whereas 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 modelled using following function.

This is a 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 the same.

As the number moves away from 0 in either direction, its membership value decreases gradually. And after some range, it vanishes to 0. The 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 the 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 the element will decrease sharply compared to the previous example, as the value of the element goes away from zero.

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

Watch on YouTube: Fuzzy set example

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Test Your Knowledge:

How would you represent the 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 a set of “good students” having a CGPA of 5.0 and above

Please Post your answers / queries / feedback in comment box below !