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 Tytuł: σ-ring and σ-algebra of Sets Autorzy: Endou, NoboruNakasho, KazuhisaShidama, Yasunari Słowa kluczowe: semiring of setsσ-ring of setsσ-algebra of sets Data wydania: 2015 Data dodania: 6-gru-201612-gru-2016 Wydawca: De Gruyter Open Źródło: Formalized Mathematics, Volume 23, Issue 1, Pages 51–57 Abstrakt: In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in [13], that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades [15]. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set [23] and field of subsets [18], respectively. In the second section, definitions of a ring and a σ-ring of sets, which are based on a semiring and a ring of sets respectively, are formalized and their related theorems are proved. In the third section, definitions of an algebra and a σ-algebra of sets, which are based on a semialgebra and an algebra of sets respectively, are formalized and their related theorems are proved. In the last section, mutual relationships between σ-ring and σ-algebra of sets are formalized and some related examples are given. The formalization is based on [15], and also referred to [9] and [16]. Afiliacja: Noboru Endou - Gifu National College of Technology, Gifu, JapanKazuhisa Nakasho - Shinshu University, Nagano, JapanYasunari Shidama - Shinshu University, Nagano, Japan URI: http://hdl.handle.net/11320/4837 DOI: 10.2478/forma-2015-0004 ISSN: 1426-26301898-9934 Typ Dokumentu: Article Występuje w kolekcji(ach): Formalized Mathematics, 2015, Volume 23, Issue 1

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