I was reading about fuzzy sets on Wikipedia. These sets are sets in which elements have a degree of membership ranging from $[0,1]$ and are defined by a set and a function mapping each element belonging to it to a degree of membership. This means that an object can either:
- Not belong to the set
- Belong to the set but have a degree of membership of $0$
- Belong to the set and have a non-zero degree of membership
Intuitively, it seems to me that the first two are equivalent, and should be merged by defining a fuzzy set either as simply a function mapping any possible object to a value, or, if that's a problem because it's domain would be the (non-existant) set of all sets, a set and a function with a co-domain of $]0,1]$.
Is there any reason fuzzy set theory was not made so that these two are equivalent? Is there any practical use to this distinction?
I could not find the definition like you wrote it on wikipedia. Maybe I did not look at the same article. Here is the definition of fuzzy sets that I know:
Let $X$ be a set. Then a fuzzy set $A$ in $X$ is characterised by a function $\mu_A : X \to [0,1]$. Here $A$ is not a subset in the classical sense, but just a name given to the set. The set itself is fully characterised by the characteristic function $\mu_A$.
Therefore your intuition is right if $\mu_A(x) = 0$ then $x$ is not an element of $A$.
It may be worth reading the original article by Zadeh: http://www.cs.berkeley.edu/~zadeh/papers/Fuzzy%20Sets-Information%20and%20Control-1965.pdf