Multiset concepts in two-universe approximation spaces

Nadya A Toumi,O A Embaby

doi:10.1186/s42787-020-00104-5

Abstract

Rough set theory over two universes is a generalization of rough set model to find accurate approximations for uncertain concepts in information systems in which uncertainty arises from existence of interrelations between the three basic sets: objects, attributes, and decisions.In this work, multisets are approximated in a crisp two-universe approximation space using binary ordinary relation and multi relation. The concept of two universe approximation is applied for defining lower and upper approximations of multisets. Properties of these approximations are investigated, and the deviations between them and corresponding notions are obtained; some counter examples are given. The suggested notions can help in the modification of the decision-making for events in which objects have repetitions such as patients visiting a doctor more than one time; an example for this case is given.

Highlights

A multiset is an unordered collection of objects in which, unlike the standard Cantorian set, the object is allowed to repeat
In 1986, multiset theory was introduced by Yager [1]
Girish and John introduced multiset topologies induced by multiset relations and the continuity between multiset topological spaces [7, 8]

Summary

Introduction

A multiset is an unordered collection of objects in which, unlike the standard Cantorian set, the object is allowed to repeat. Approximation of multisets in crisp approximation space Definition 3.1 Let U and V be two finite non-empty universes of discourse and R ∈ P(U × V) be a binary relation from U to V. Proposition 3.1 In a two-universe model (U, V, R) with the binary relation R, the approximation operators RP and RP A2 ∈ [U]w: ðL1Þ RPðAÞ 1⁄4 ðRPðAcÞÞc: ðL3Þ RPðA1∩A2Þ 1⁄4 RPðA1Þ∩RPðA2Þ: ððUL52ÞÞAR1⊆PðAφ2Þ⟹ 1⁄4 φR:PðA1Þ⊆RPðA2Þ: ðU4Þ RPðA1∩A2Þ⊆RPðA1Þ∩RPðA2Þ: satisfy the following properties for all A, A1, ðL2Þ RPðUÞ 1⁄4 V : ðððULU413ÞÞÞRRRPðPPAððAA1∪1Þ∪A1⁄4A2Þ2ð⊇ÞRR1⁄4PPððARAcPÞ1ðÞÞAc∪:1RÞP∪ðRAP2ðÞA: 2Þ: . Remark 3.1 If R ∈ P(U × V) is a serial relation in a two-universe approximation space (U, V, R), the properties L6, U6, and LU are not true in general, as shown in the following example: Example 3.1 Let U = {a1, a2, a3, a4}, V = {b1, b2, b3, b4, b5}, and R be a binary relation from U to V defined as:.

C R P ðRP ðAÞÞ ðaÞ

Conclusion and future work