Pub Date : 2018-10-03DOI: 10.1201/9781315273761-23
Tom Kelliher
1. Present the following explanation to students: Consider the function x x h 4 ) ( . Using an input of 9 to evaluate h, we see that . 6 36 ) 9 ( 4 ) 9 ( h So, h(9) = 6. Since we performed two operations—multiplication and finding the square root—we can think of h as a composite of two functions. Let’s call these two functions f(x) and g(x), with f(x) = 4x and g(x) = x . We can evaluate h(9) by finding the output of f(9) and using that output as the input of function g. First, use 9 as the input for f, and find f(9) = 4(9) = 36. Next, use that output as the input for g and find 6 36 ) 36 ( g . Therefore, 6 ) 36 ( )] 9 ( [ ) 9 ( ) 9 ( g f g f g h . When we use an output of one function as an input for another function, we are creating a composition of functions. In our example, h is a composition of f and g, which is written as ) ( ) ( or )] ( [ ) ( x f g x h x f g x h . Both equations are read “h(x) equals g of f of x.”
1. 给学生们讲解如下:考虑函数x x h 4。用输入9求h的值，我们看到了。6 36) 9 (4) 9 (h所以h(9) = 6。因为我们执行了两个操作——乘法和求平方根——我们可以把h看作两个函数的复合。我们称这两个函数为f(x)和g(x)， f(x) = 4x, g(x) = x。我们可以通过找到f(9)的输出并使用该输出作为函数g的输入来计算h(9)。首先，使用9作为f的输入，并找到f(9) = 4(9) = 36。接下来，使用该输出作为g的输入，并找到6 36)36 (g。因此，6)36 ()]9 ([)9 ()9 (g g g g h。当我们使用一个函数的输出作为另一个函数的输入时，我们正在创建一个函数的组合。在我们的例子中，h是f和g的组合，写成)()(或)]([)(x f g x h x f g x h。两个方程都是h(x) = g (f (x))
{"title":"Composition of Functions","authors":"Tom Kelliher","doi":"10.1201/9781315273761-23","DOIUrl":"https://doi.org/10.1201/9781315273761-23","url":null,"abstract":"1. Present the following explanation to students: Consider the function x x h 4 ) ( . Using an input of 9 to evaluate h, we see that . 6 36 ) 9 ( 4 ) 9 ( h So, h(9) = 6. Since we performed two operations—multiplication and finding the square root—we can think of h as a composite of two functions. Let’s call these two functions f(x) and g(x), with f(x) = 4x and g(x) = x . We can evaluate h(9) by finding the output of f(9) and using that output as the input of function g. First, use 9 as the input for f, and find f(9) = 4(9) = 36. Next, use that output as the input for g and find 6 36 ) 36 ( g . Therefore, 6 ) 36 ( )] 9 ( [ ) 9 ( ) 9 ( g f g f g h . When we use an output of one function as an input for another function, we are creating a composition of functions. In our example, h is a composition of f and g, which is written as ) ( ) ( or )] ( [ ) ( x f g x h x f g x h . Both equations are read “h(x) equals g of f of x.”","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"27 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":"126044880","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: 10.1201/9781315273761-34
{"title":"Denumerable and Countable Sets","authors":"","doi":"10.1201/9781315273761-34","DOIUrl":"https://doi.org/10.1201/9781315273761-34","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"11 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":"124049184","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: 10.1201/9781315273761-31
K. E. Hummel
{"title":"The Real Number System","authors":"K. E. Hummel","doi":"10.1201/9781315273761-31","DOIUrl":"https://doi.org/10.1201/9781315273761-31","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"151 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":"132799243","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":"Quantifiers and Predicates Chapter","authors":"","doi":"10.1201/9781315273761-9","DOIUrl":"https://doi.org/10.1201/9781315273761-9","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"14 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":"133130760","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: 10.1201/9781315273761-27
{"title":"The Systems of Whole and Natural Numbers","authors":"","doi":"10.1201/9781315273761-27","DOIUrl":"https://doi.org/10.1201/9781315273761-27","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"5 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":"133833193","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}
Propositional logic was eventually refined using symbolic logic. The 17th/18th century philosopher Gottfried Leibniz (an inventor of calculus) has been credited with being the founder of symbolic logic. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were re-achieved by logicians like George Boole and Augustus De Morgan in the 19th century completely independent of Leibniz.
命题逻辑最终通过符号逻辑得到完善。17 /18世纪的哲学家戈特弗里德·莱布尼茨(微积分的发明者)被认为是符号逻辑的创始人。虽然他的工作是同类研究中的第一个，但在更大的逻辑学界并不为人所知。因此，莱布尼茨取得的许多进步，在19世纪被乔治·布尔(George Boole)和奥古斯都·德·摩根(Augustus De Morgan)等完全独立于莱布尼茨的逻辑学家重新实现。
{"title":"Logic and Propositional Calculus","authors":"","doi":"10.1201/9781315273761-7","DOIUrl":"https://doi.org/10.1201/9781315273761-7","url":null,"abstract":"Propositional logic was eventually refined using symbolic logic. The 17th/18th century philosopher Gottfried Leibniz (an inventor of calculus) has been credited with being the founder of symbolic logic. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were re-achieved by logicians like George Boole and Augustus De Morgan in the 19th century completely independent of Leibniz.","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":"117060803","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: 10.1201/9781315273761-24
{"title":"Image and Preimage Functions","authors":"","doi":"10.1201/9781315273761-24","DOIUrl":"https://doi.org/10.1201/9781315273761-24","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"73 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114092146","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: 10.1201/9781315273761-17
{"title":"Generalized Union and Intersection","authors":"","doi":"10.1201/9781315273761-17","DOIUrl":"https://doi.org/10.1201/9781315273761-17","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"35 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":"133281846","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: 10.1201/9781315273761-39
E. G. Powerset
{"title":"Least Upper Bound and Greatest Lower Bound","authors":"E. G. Powerset","doi":"10.1201/9781315273761-39","DOIUrl":"https://doi.org/10.1201/9781315273761-39","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"70 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":"130459212","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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