{"title":"Sets and Counting","authors":"P. Scott","doi":"10.1002/9781119563549.ch2","DOIUrl":"https://doi.org/10.1002/9781119563549.ch2","url":null,"abstract":"A detailed explanation of sets and counting, including subsets, venn diagrams, cardinality, transfinite numbers with problems and their solutions are provided.","PeriodicalId":258847,"journal":{"name":"Advanced Mathematics","volume":"183 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129652684","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":"Relations","authors":"Michael Lambert","doi":"10.1017/9781108539890.006","DOIUrl":"https://doi.org/10.1017/9781108539890.006","url":null,"abstract":". A ‘double category of relations’ is defined in this paper as a cartesian equipment in which every object is suitably discrete. The main result is a characterization theorem that a ‘double category of relations’ is equivalent to a double category of relations on a regular category when it has strong and monic tabulators and a double-categorical subobject comprehension scheme. This result is based in part on the recent characterization of double categories of spans due to Aleiferi. The overall development can be viewed as a double-categorical version of that of the notion of a “functionally complete bicategory of relations” or a “tabular allegory.”","PeriodicalId":258847,"journal":{"name":"Advanced Mathematics","volume":"8 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":"130223526","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":"Logic and Proofs","authors":"C. Reis, S. Rankin","doi":"10.1142/9789814335652_0001","DOIUrl":"https://doi.org/10.1142/9789814335652_0001","url":null,"abstract":"• A proposition is a declarative statement (sentence) that is true or false. For example, “Mars is a planet” is a proposition that is true, and “Jupiter is a star” is a proposition that is false. Questions, commands and exclamations are examples of sentences that are not propositions. • Truth values: They are the values “true” (denoted T) and “false” (denoted F). • A proposition that conveys a single fact is called a simple proposition. • Simple propositions can be combined and/or negated, using so-called logical connectives, to form more elaborate propositions called compound propositions. • Logical connectives: They are the logical operations “and”, “or”, and “not”. The first two are binary operations, that is, they take two operands where each operand is a proposition. The “not” is a unary operation in that it takes a single operation. • Examples of compound propositions: o Mars is a planet and Jupiter is a star o Mars is a planet or Jupiter is a star o not (Jupiter is a star). In plain English, this statement means “Jupiter is not a star”. o Mars is a planet and (Jupiter is a star or not(Pluto is a comet)) • Notations: o The operation “and” is denoted ∧. (In Boolean algebra, it is denoted by the dot “.” ). o The operation “or” is denoted ∨. (In Boolean algebra, it is denoted by + ). o The operation “not” is denoted ¬, ~, ̅(overbar), or ′. • Logic is concerned much more about the truth values of propositions than about their meanings. • Therefore, and for convenience and efficiency, propositions are often denoted with single-letter symbols such as p, q, r, etc. And the focus is on whether a proposition p is true or false, rather than on what p means/designates. For example, we can denote “Mars is a planet” by p, “Jupiter is a star” by q, and “Pluto is a comet” by r. Then the above examples of compound statements can be succinctly represented as: o “Mars is a planet and Jupiter is a star”: pp ∧ qq. This (i.e., pp ∧ qq) is called a conjunction. o “Mars is a planet or Jupiter is a star”: pp ∨ qq. This (i.e., pp ∨ qq) is called a disjunction. o “not (Jupiter is a star)”: ¬pp, ~pp, ?̅?p, or pp′.","PeriodicalId":258847,"journal":{"name":"Advanced Mathematics","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128519629","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 Real and Complex Number Systems","authors":"E. R. Bennett","doi":"10.1002/9781119563549.ch4","DOIUrl":"https://doi.org/10.1002/9781119563549.ch4","url":null,"abstract":"","PeriodicalId":258847,"journal":{"name":"Advanced Mathematics","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126316709","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}
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