jrh7328 – Proofs in the Pudding

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

June 1, 2023jrh7328 1 Comment

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.

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.… Read the rest