Some Properties for the Restriction of P^*-field of Sets

Authors

Hind Abbas Department of Mathematics / College of Computer Science and Mathematics / Tikrit University/ Iraq.
hassan Ebrahim Department of Mathematics / College of Computer Science and Mathematics / Tikrit University/ Iraq.
Ali Al-Fayadh Department of Mathematics and Computer Applications / College of Science /Al – Nahrain University/ Iraq

DOI:

https://doi.org/10.30526/35.3.2814

Keywords:

σ-field, σ– ring, field, smallest σ-field, restriction

Abstract

The restriction concept is a basic feature in the field of measure theory and has many important properties. This article introduces the notion of restriction of a non-empty class of subset of the power set on a nonempty subset of a universal set. Characterization and examples of the proposed concept are given, and several properties of restriction are investigated. Furthermore, the relation between the P*–field and the restriction of the P*–field is studied, explaining that the restriction of the P*–field is a P*–field too. In addition, it has been shown that the restriction of the P*–field is not necessarily contained in the P*–field, and the converse is true. We provide a necessary condition for the P*–field to obtain that the restriction of the P*–field is included in the P*–field. Finally, this article aims to study the restriction notion and give some propositions, lemmas, and theorems related to the proposed concept.

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Abbas, H.F. ; Ebrahim, H.H. ; Al-Fayadh, A. P^*–Field of sets and Some of its Properties, Accepted in Computers and Mathematics with Applications, 2022.