{"title":"Counting and the Size of Sets","authors":"","doi":"10.1201/9781315275444-17","DOIUrl":"https://doi.org/10.1201/9781315275444-17","url":null,"abstract":"","PeriodicalId":371419,"journal":{"name":"Fundamentals of Abstract Analysis","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122873705","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":"The Construction of the Real Numbers","authors":"Holmes","doi":"10.1201/9781315275444-15","DOIUrl":"https://doi.org/10.1201/9781315275444-15","url":null,"abstract":"Commutative laws: Addition and multiplication are commutative operations. Associative laws: Addition and multiplication are associative operations. Distributive law: x(y + z) = xy + xz as usual. Identity for multiplication: 1 is the identity element for multiplication. Inverse law for multiplication: Every number a has a multiplicative inverse a. (recall that zero is not in our system). Definition: We define x < y as ∃z(x + z = y) then >, ≤, ≥ are defined using < as usual. Trichotomy 1: For any x and y, either x < y, y < x, or x = y. Trichotomy 2: For any x and y, either ¬x < y or ¬y < x.","PeriodicalId":371419,"journal":{"name":"Fundamentals of Abstract Analysis","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121243355","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":"Sums and Products","authors":"C. Pomerance, A. Schinzel","doi":"10.1201/9781315275444-19","DOIUrl":"https://doi.org/10.1201/9781315275444-19","url":null,"abstract":"What could be simpler than to study sums and products of integers? Well maybe it is not so simple since there is a major unsolved problem: For any positive and arbitrarily large numbers N, is there a set of N positive integers where the number of pairwise sums is at most N2− and likewise, the number of pairwise products is at most N2−? Erdös and Szemerédi conjecture no. This talk is directed at another problem concerning sums and products, namely how dense can a set of positive integers be if it contains none of its pairwise sums and products? For example, take the numbers that are 2 or 3 mod 5, a set with density 2/5. Can you do better? This talk reports on recent joint work with P. Kurlberg and J.C. Lagarias. Sums and Products Carl Pomerance Dartmouth College FEATURED LECTURE BY: Marc-Thorsten Hütt","PeriodicalId":371419,"journal":{"name":"Fundamentals of Abstract Analysis","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128399597","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}