Near Sets: An Introduction

This article gives a brief overview of near sets. The proposed approach in introducing near sets is to consider a set theory-based form of nearness (proximity) called discrete proximity. There are two basic types of near sets, namely, spatially near sets and descriptively near sets. By endowing a nonempty set with some form of a nearness (proximity) relation, we obtain a structured set called a proximity spaces. Let $${\mathcal{P}(X)}$$ denote the set of all subsets of a nonempty set X. One of the oldest forms of nearness relations p (later denoted by δ) was introduced by E. Čech during the mid-1930s, which leads to the discovery of spatially near sets, i.e., those sets that have elements in common. That is, given a proximity space (X, δ), for any subset $${A \in \mathcal{P}(X)}$$ , one can discover nonempty nearness collections $${\xi(A) = \{B \in \mathcal{P}(X): A \, \delta \, B\} }$$ . Recently, descriptively near sets were introduced as a means of solving classification and pattern recognition problems arising from disjoint sets (i.e, sets with empty spatial intersections) that resemble each other. One discovers descriptively near sets by choosing a set of probe functions Φ that represent features of points in a set and endowing the set of points with a descriptive proximity relation δ Φ and obtaining a descriptively structured set (called descriptive proximity space). Given a descriptive proximity spaces (X, δ Φ), one can discover collections of subsets that resemble each other. This leads to the discovery of descriptive nearness collections $${\xi_{\Phi}(A) = \{B \in \mathcal{P}(X): A \,\delta_{\Phi} \, B\} }$$ . That is, if $${B \in \xi_{\Phi}(A)}$$ , then A δ Φ B (relative to the chosen features of points in X, A resembles B). The focus of this tutorial is on descriptively near sets.

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