Can We Detect Crisp Sets Based Only on the Subsethood Ordering of Fuzzy Sets? Fuzzy Sets and/or Crisp Sets Based on Subsethood of Interval-Valued Fuzzy Sets?

Fuzzy sets are naturally ordered by the subsethood relation A ⊆ B. If we only know which set which fuzzy set is a subset of which – and have no access to the actual values of the corresponding membership functions – can we detect which fuzzy sets are crisp? In this paper, we show that this is indeed possible. We also show that if we start with interval-valued fuzzy sets, then we can similarly detect type-1 fuzzy sets and crisp sets. 1 Formulation of the Problem Fuzzy sets: a brief reminder. A fuzzy set is usually defined as a function μ :U → [0,1] from some set U (called Universe of discourse) to the interval [0,1]; see, e.g., [1, 2, 3]. This function is also known as a membership function. A fuzzy set A with a membership function μA(x) is called a subset of a fuzzy set B with a membership function μB(x) if μA(x)≤ μB(x) for all x. The subsethood relation is an order in the sense that it is reflexive (A ⊆ A), asymmetric (A ⊆ B and B⊆ A imply A= B), and transitive (A⊆ B and B⊆C imply A⊆C). Traditional (crisp) sets S can be viewed as particular cases of fuzzy sets, with their characteristic functions playing the role of membership functions: μS(x) = 1 if x ∈ S and μS(x) = 0 if x ̸∈ S. A natural question: can we detect crisp sets based only on the subsethood ordering of fuzzy sets? If we have a class F of all fuzzy sets, and for each fuzzy Christian Servin Computer Science and Information Technology Systems Department, El Paso Community College 919 Hunter, El Paso, Texas 79915, USA, [email protected] Gerardo Muela and Vladik Kreinovich Department of Computer Science, University of Texas at El Paso, 500 W. University El Paso, Texas 79968, USA, e-mail: [email protected], [email protected]