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http://hdl.handle.net/11320/4837
Tytuł: | σ-ring and σ-algebra of Sets |
Autorzy: | Endou, Noboru Nakasho, Kazuhisa Shidama, Yasunari |
Słowa kluczowe: | semiring of sets σ-ring of sets σ-algebra of sets |
Data wydania: | 2015 |
Data dodania: | 6-gru-2016 12-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, Japan Kazuhisa Nakasho - Shinshu University, Nagano, Japan Yasunari Shidama - Shinshu University, Nagano, Japan |
URI: | http://hdl.handle.net/11320/4837 |
DOI: | 10.2478/forma-2015-0004 |
ISSN: | 1426-2630 1898-9934 |
Typ Dokumentu: | Article |
Występuje w kolekcji(ach): | Formalized Mathematics, 2015, Volume 23, Issue 1 |
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