Fuzzy Set Theory: To Be in a Set, or not Be in a Set? Maybe It’s not that Simple…

John Hileman

When we began this class, we all learned one thing about sets: an element can either be in a set or not. In our proofs, $x \in A$ and $x \not\in A$ was a contradictory statement. However, fuzzy set theory makes that a statement that is entirely possible. Fuzzy set theory, as a philosophical concept, was introduced in the early 1900s by Zadeh and Goguen when they declared that the concept of traditional set theory does not make sense in a real-world application because in real-life, things are not nearly as black or white. In reality, there is a lot more gray area, which is what fuzzy set theory as a whole is trying to conceptualize.

Fuzzy Set Theory

The figure above shows a generalized example to introduce you to the concept of fuzzy set theory. Fuzzy set theory could take an entire course to explain, and there are many definitions and proofs involved. However, as a basic introduction for you all, I will talk about some definitions that strongly relate to our course. This leads us to our first definition, a fuzzy set is one of the form $A^{\sim} = \{(x, \mu_{\sim}(x)) \mid x \in X\}$ where $X$ is a traditional set, and $\mu_{\sim}$ is a fuzzy membership function that maps $X$ to a membership space that has a value between 0 and 1. Think of this like the degree of the membership (near 1 is strong membership). So in order to define a traditionally valued set using fuzzy set theory, a set A could theoretically be defined by $A = \{x \in A^{\sim} \mid \mu_{\sim}(x) = 1\}$ .

After defining fuzzy sets, we can now do some translations from fuzzy set theory into our concepts of traditional set theory. To begin with, we have the notions of fuzzy set operations. I will define some familiar ones like unions and intersections. I will preface this by saying that when defining set operations, it is typical to define the set operation of the membership function and not the actual sets. The membership function for the union of fuzzy sets is denoted by $\mu_{A^{\sim} \cup B^{\sim}} = \min(\mu_{A_{\sim}}, \mu_{B^{\sim}})$ . The membership function for the intersection of fuzzy sets is denoted by $\mu_{A^{\sim} \cap B^{\sim}} = \max(\mu_{A_{\sim}}, \mu_{B^{\sim}})$ . The first component of the pair $(x, \mu_{\sim}(x))$ should be unioned or intersected in the traditional sense. This intuitively makes sense because intersections are smaller than unions, even in this new fuzzy logic. Speaking of “smaller than,” we can also define new cardinality. The cardinality of a finite fuzzy set is defined by $|A^{\sim}| = \sum_{x \in X} \mu_{A^{\sim}}(x)$ . Note that this sum may or may not diverge for infinite fuzzy set.

Another concept that we learned about regarding traditional set theory that can be defined in fuzzy set theory as well are relations. A fuzzy relation is defined by $R^{\sim} = \{(x,y), \mu_{R^{\sim}}(x,y) \mid (x,y) \subseteq X \times Y\}$ . However, instead of something strictly defined as we did in relations, a relation could be something like “considerably larger than.”

Now, you may ask, why would I care about these fuzzy sets? Well, fuzzy sets model real life much more accurately. Someone can be very old or very young, not exactly one age which is less useful. Fuzzy set theory has interesting applications to machine learning, artificial intelligence, economics, and other business aspects. The example below is a fuzzy controller involved in data mining for example.

Fuzzy Controller

In conclusion, fuzzy set theory is a new way to define what we spent all quarter learning about, but it goes more in-depth and explains things that model real-life in a much better way. This is used in artificial intelligence especially as a better way that the computer can model real-life simulations. Thank you for your time in reading this article about fuzzy set theory, and for further reading consult my source below.

Sources

Lehmann, I., Weber, R. & Zimmermann, H.J. Fuzzy set theory. OR Spektrum 14, 1–9 (1992). https://doi.org/10.1007/BF01783496.

One Comment


June 3, 2023
mbm9123

This is a great post John! It is interesting to think about the degree to which an element is in a set. I really like how you not only gave an overview of what fuzzy sets are, but also described their importance in the real world. I really like how you used the example of ages and I was thinking about what elements would have degree of membership 0 or 1. For example, would the set of people aged 200 have a degree of membership equal to 0 since nobody is that old? You also briefly mentioned the cardinality of fuzzy sets, which makes we wonder if the theorems we learned about infinite, countably infinite, and uncountable still apply to fuzzy sets? I’m thinking that even the definitions of countable and uncountable sets would be different in fuzzy set theory. You also contrast fuzzy sets with the sets we learned about in class, which made me think about if the sets from class can be classified as fuzzy sets as well as long as the fuzzy membership function maps only to 0 or 1 and nothing in between? This post was well-written and makes me know want to look more into fuzzy sets, their theorems, and their applications.!

Fuzzy Set Theory: To Be in a Set, or not Be in a Set? Maybe It’s not that Simple…

Sources

jrh7328

One Comment

Leave a Reply Cancel reply