A Min-transitive Fuzzy Left-Relationship on Finite Sets Based on Distance to Left

Abstract

We first consider the “discrete” domain of non-empty finite subsets $X = \{x_1, x_2, \ldots , x_m \}$ of $\mathcal {R} = (-\infty , +\infty )$, where each $x_i$ has a weight or probability $P_X(x_i) > 0$. We define a min-transitive fuzzy left-relationship $\varLambda _D(X, Y)$ on this domain, which accounts for the distance to the left L(x, y) = max(0, $y-x$) between points $x \in X$ and $y \in Y$. Then, we extend $\varLambda _D(\cdot , \cdot )$ to finite intervals $X \subset \mathcal {R}$ with arbitrary probability density function $p_{ _X}\!(x)$ on them, and to fuzzy sets X with finite support, which may be a discrete set or an interval, and membership function $\mu _{ _X}\!(x)$.