{"title":"Transfinite Cardinal Numbers","authors":"J. N. Salunke, B. S. Satpute","doi":"10.1201/9781315273761-36","DOIUrl":"https://doi.org/10.1201/9781315273761-36","url":null,"abstract":"For transfinite cardinal number α, using Zorn’s lemma we have given a simple proof which is understandable to undergraduate students, of the result α + α = αα = α, that is, idempotency for addition and multiplication. Moreover for a cardinal number with 2 ≤ < α we obtain easily α + = α = α, α = α α = 2 α , α < . Using these results we get many results directly as א0 + ∁ = א0∁ = ∁ + ∁ = ∁∁ = ∁, א0 א0 = ∁ = ∁0 = 20 , ∁ = א0 ∁ = 2 where א0 = card N, ∁ = card R. -------------------------------------------------------------------------------------------------------------------------------------Date of Submission: 05-08-2018 Date of acceptance: 22-08-2018 -------------------------------------------------------------------------------------------------------------------------------------","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132389421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tautologies and Validity Chapter","authors":"","doi":"10.1201/9781315273761-8","DOIUrl":"https://doi.org/10.1201/9781315273761-8","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126107090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}