Pub Date : 2018-10-03DOI: 10.1201/9781315273761-29
{"title":"The System Q of Rational Numbers","authors":"","doi":"10.1201/9781315273761-29","DOIUrl":"https://doi.org/10.1201/9781315273761-29","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"1 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":"131292347","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}
Pub Date : 2018-10-03DOI: 10.1201/9781315273761-30
{"title":"Other Aspects of Order","authors":"","doi":"10.1201/9781315273761-30","DOIUrl":"https://doi.org/10.1201/9781315273761-30","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"23 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":"133615491","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}
Pub Date : 2018-10-03DOI: 10.1201/9781315273761-28
{"title":"The System Z of Integers","authors":"","doi":"10.1201/9781315273761-28","DOIUrl":"https://doi.org/10.1201/9781315273761-28","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"40 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":"125341125","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}
Pub Date : 2018-10-03DOI: 10.1201/9781315273761-26
Czes Law Byli´nski
In this paper we define binary and unary operations on domains. number of schemes useful in justifying the existence of the operations are proved. The articles [3], [1], and [2] provide the notation and terminology for this paper. The arguments of the notions defined below are the following: f which is an object of the type Function; a, b which are objects of the type Any. The functor f .(a, b), with values of the type Any, is defined by it = f .a, b. One can prove the following proposition (1) for f being Function for a,b being Any holds f .(a, b) = f .a, b. In the sequel A, B, C will denote objects of the type DOMAIN. The arguments of the notions defined below are the following: A, B, C which are objects of the type reserved above; f which is an object of the type Function of [:A, B:], C; a which is an object of the type Element of A; b which is an object of the type Element of B. Let us note that it makes sense to consider the following functor on a restricted area. Then f .(a, b) is Element of C.
本文定义了定义域上的二元和一元运算。证明了若干有助于证明运算存在性的方案。文章[3]、[1]和[2]提供了本文的符号和术语。下面定义的概念的实参如下:f是函数类型的对象;a, b，它们是Any类型的对象。具有Any类型值的函子f .(a, b)由它= f .a, b来定义。可以证明下列命题(1):f为a,b为Any的函数，使f .(a, b) = f .a, b成立。在续集a,b, C表示类型为DOMAIN的对象。下面定义的概念的实参如下:A, B, C，它们是上述保留类型的对象;f是类型为Function of [:A, B:]， C;a是a的要素类型的对象;b是b的Element类型的对象，让我们注意到在受限区域上考虑以下函子是有意义的。那么f (a, b)是C的元素。
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Pub Date : 2018-10-03DOI: 10.1201/9781315273761-15
Tim Pilachowski
Subsets and predicates. For these notes we'll look at one set U , called the universal set, and its subsets. In later notes, we'll build new sets out of old ones using the product construction and the powerset construction. The universal set U corresponds to the domain of discourse in predicate logic when we're only considering unary predicates on that domain. Let, for instance, U be the set of integers, usually denoted Z. A couple of unary predicates for this domain are S(x): x is a perfect square," and P (x): x is a positive integer." These two predicates correspond to two subsets of U . The rst corresponds to the set of perfect squares which includes 0; 1; 4; 9; etc., and the second corresponds to the set of positive integers which includes 1; 2; 3; etc. The subset that corresponds to a unary predicate is called the extent of the predicate. There's such a close correspondence between a unary predicate and it's extent that we might as well use the same symbol for both. So, we can use S for the subset of perfect squares, or S for the predicate which indicates with the notation S(x) whether an integer x is a perfect square or not. There are a couple of ways to use notation to specify a set. One is by listing its elements, at least the rst few, and hoping the reader can understand your intent.
{"title":"Sets and Set Operations","authors":"Tim Pilachowski","doi":"10.1201/9781315273761-15","DOIUrl":"https://doi.org/10.1201/9781315273761-15","url":null,"abstract":"Subsets and predicates. For these notes we'll look at one set U , called the universal set, and its subsets. In later notes, we'll build new sets out of old ones using the product construction and the powerset construction. The universal set U corresponds to the domain of discourse in predicate logic when we're only considering unary predicates on that domain. Let, for instance, U be the set of integers, usually denoted Z. A couple of unary predicates for this domain are S(x): x is a perfect square,\" and P (x): x is a positive integer.\" These two predicates correspond to two subsets of U . The rst corresponds to the set of perfect squares which includes 0; 1; 4; 9; etc., and the second corresponds to the set of positive integers which includes 1; 2; 3; etc. The subset that corresponds to a unary predicate is called the extent of the predicate. There's such a close correspondence between a unary predicate and it's extent that we might as well use the same symbol for both. So, we can use S for the subset of perfect squares, or S for the predicate which indicates with the notation S(x) whether an integer x is a perfect square or not. There are a couple of ways to use notation to specify a set. One is by listing its elements, at least the rst few, and hoping the reader can understand your intent.","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"71 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":"114208717","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}
Pub Date : 2018-10-03DOI: 10.1201/9781315273761-10
{"title":"Techniques of Derivation and Rules of Inference","authors":"","doi":"10.1201/9781315273761-10","DOIUrl":"https://doi.org/10.1201/9781315273761-10","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"219 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":"130419478","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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