Given a fuzzy set, find the level set

Question

If you are give a fuzzy set

$A = \frac{0.1}{x_{1}}+ \frac{0.3}{x_{2}} + \frac{0.6}{x_{3}} + \frac{1}{x_{4}}$

How do you find the level set?

Answer 1

For any $a\in L$, an $a$-level set is a subset of $X$ defined as $X_a:=\{x\in X: A(x)\geq a\}$, where $A(x)\in L$ denotes the membership value of $x$ in your fuzzy set $A$. In your case (assuming here $L=[0,1]$), you would have, for example: $X_{0.2}=\{x_2,x_3,x_4\}$ and $X_{0.5}=\{x_3,x_4\}$. You can do this for any value in the unit interval. What will $X_0$ always be equal to?

Answer 2

Klir & Folger [1] say that $\alpha$-cut of a fuzzy set $A$ is a crisp set $A_\alpha$ that contains the elements that have a membership grade in $A$ greater than or equal to $\alpha$.

And the set of all levels $\alpha \in [0,1]$ that represent distinct $\alpha$-cuts is called a level set of $A$. That would be $$ \Lambda_A = \{ \alpha\ |\ \mu_A(x) = \alpha \text{ for some } x\in X\}. $$

The $\alpha$-cut here is same as $\alpha$-level set in massy255's answer.

[1] Klir & Folger, Fuzzy Sets, Uncertainty and Information, 1988.