proofs: a long form mathematics textbook pdf

Translate Bates College has a great introduction to LaTeX, packed with templates and lessons. The writing is relaxed and conversational, and includes periodic attempts at humor.This text is also an introduction to higher mathematics. Use the first one that responds. It is always crystal clear why each idea, term, and technique was presented in the place where it was. return; } annas-archive.org A decade ago there is no way that Springer would approach someone like me to write an analysis textbook, however today Amazon's self-publishing arm couldn't be happier to print my book. polski [pl] I am a 2017 Project NExT Beautiful Blue dot. A "file MD5" is a hash that gets computed from the file contents, and is reasonably unique based on that content. The first appendix contains a construction the reals, and the second is a collection of additional peculiar and pathological examples from analysis.The author believes most textbooks are extremely overpriced and endeavors to help change this. If you would rather learn in a language other than English, check out learnlatex.org. The metadata might differ for the different libraries, even when one library initially just copied the metadata from another one, since contributors of the different libraries can subsequently change the metadata independently. It could add to the students experience if there were more complete and informative indices and glossary. This book will initiate you into an esoteric world. I am passionate about my teaching and conduct research in several areas of combinatorics. Obsolescence is not really an issue with this material. The text aims to make the ideas visible, and contains over 200 illustrations. They include finer comments on the material, study tips, historical notes, comments on mathematical culture, and more. to find the book on Amazon. read more. This book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. Most of what I've learned has been through troubleshooting with Google, which usually directs me to some page on tex.stackexchange.com in which someone many years ago asked the same thing. However, this is not done in an inappropriate way nor is prior knowledge of anything about Canada required at any point in any way. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. However, some professors who use the book don't want any solutions posted. % } They include finer comments on the material, study tips, historical notes, comments on mathematical culture, and more. well, this doesn't push into measures either, but during those first weeks when you're beating your head against convergences and continuity, this would have been a wonderful guide (at significant cost savings, natch). Each chapter ends with exercises and an open question, as well as ``pro-tips,'' which are short thoughts on things I wish I had known when I took my intro-to-proofs class. Instead proof by induction is at the end of the text. [el] That pedagogical strategy is one of the strengths of this book. Proofs: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series): Cummings, Jay: 9798595265973: Amazon.com: Books Books Science & Math Mathematics Buy new: $16.56 List Price: $17.50 Save: $0.94 (5%) Get Fast, Free Shipping with Amazon Prime FREE Returns FREE delivery Friday, March 3 if you spend $25 on items shipped by Amazon read more. document.querySelector(".js-annas-archive-gs").href = loc.replace(currentDomainToReplace, "annas-archive.gs"); Zoom Link: https://csus.zoom.us/j/3703467635, E-mail: Jay.Cummings [ampersat] csus [dot] edu. There were no issues related to interface. The writing is relaxed and conversational, and includes periodic attempts at humor. Also, after each chapter's exercises is an introduction to an unsolved problem in mathematics.In the first appendix we discuss some further proof methods, the second appendix is a collection of particularly beautiful proofs, and the third is some writing advice. Another good reference site is https://en.wikibooks.org/wiki/LaTeX. This is not to say that everything is explained in exactly one way: there is a great feature in this book, particularly in the first half heavy on logic and proofs, of explaining many ideas both in quite formal, mathematical ways and also in clear and precise but more natural-language sounding ways. Reviewed by Nic Gilbertson, Assistant Professor, Winona State University on 2/1/18, This is a well-written text, that can be readily used for introduction to proofs and logic course at the undergraduate level. [ja] I More information can be found in the Book(s) tab on the left or on its page on Amazon . Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. This proofs has even more scratch work and proof sketches. The book gives a beautiful, complete, and careful exposition of its central material -- logic and proofs -- and of several beautiful and powerful applications in different areas of mathematics. function checkOtherDomains() { This book is absolutely superb for a one-semester bridge course to the advanced mathematics curriculum. As such, it would do students an enormous service by giving them a beautiful and clear introduction to how mathematicians really think and write about mathematics. I may even cover too much logic before getting into the more mathematical applications and proof writing. One difference from other books of its type is that the text on proof y induction is not with the other As a compromise, I chose 10 problems each chapter and gave a complete solution to those problems. This is done in-part through the chosen examples and theorems. It covers everything one would want for an introduction to proofs and abstraction, but I think it might take too long to get to the "meaty" subjects. That said, I believe a shift is beginning to occur in the textbook market. espaol [es] We merely link to other peoples websites. The text aims to make the ideas visible, and contains over 200 illustrations. ), it gives some non-mathematical ("real world") examples of objects and reasoning to help make the material more intuitive and less intimidating for beginners. English [en] The author mentions other widely used alternate notations when introducing new notation. } Furthermore, following every chapter is an introduction to an area of math. Examples often drive the narrative and challenge the intuition of the reader. The proofs are not terse, and aim for understanding over economy. + Math.random(); slovenina [sk] (function() { el.action = el.action.replace(currentDomainToReplace, domain); There does no appear to be any interface issues. } for just $16, and in my own classes I offer a free alternative. This book is a very comprehensive look at proof methods. I also believe that learning mathematics has become too expensive and commercialized, and textbooks in particular are far overpriced. Also, after each chapter's exercises is an introduction to an unsolved problem in mathematics.In the first appendix we discuss some further proof methods, the second appendix is a collection of particularly beautiful proofs, and the third is some writing advice. For this book and the proofs book, the writing is far smore relaxed than most textbooks, and I have included periodic historical notes, poor attempts at humor, and occasional diversions into other interesting areas of mathematics. All document files are the property of their respective owners, please respect the publisher and the author for their copyrighted creations. The book gives a beautiful, complete, and careful exposition of its central material -- logic and proofs -- and of several beautiful and powerful applications in different areas of mathematics. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. This 500-page book is currently available onAmazon for just $16, and can be, I have argued that there is a need for a new style of textbook, and fortunately I came to this realization just as the gatekeepers of publishing were critically weakened by the advent of self-publishing and online marketplaces. // Possible domains we can encounter: Copyright 2019 Long-Form Math - All Rights Reserved. The examples used range from mathematical and non-mathematical examples. if (! %PDF-1.5 This is a well-written text, that can be readily used for introduction to proofs and logic course at the undergraduate level. All download options should be safe to use. This text is also an introduction to higher math. Proofs: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series) (January 19, 2021) by Jay Cummings (Author) 4.8 out of 5 stars (347) This textbook is designed for students. Jay Cummings. } It may be written in 'long-form,' style but the time you spend reading more words than typical for a math book, you more than save in understanding concepts on your first or second read through (rather as opposed to staring at a sentence or equation 10 times in a row and failing to understand what's going). // Check if there are other domains that are still up. The text is about as modular as it is possible to imagine for this material -- so that, e.g., it is hard to imagine doing first order logic without doing propositional logic first, so some version of Chapter 1 has to be done *before* Chapter 4. } The writing is relaxed and conversational, and includes periodic attempts at humor.This text is also an introduction to higher mathematics. document.querySelector(".js-annas-archive-org").href = loc.replace(currentDomainToReplace, "annas-archive.org"); i would have really appreciated this book back when i underwent "Trial by Fire" aka Analysis I. bartle's unassuming blue volume as used by GT was actually pretty solid prose, but the exercises were lacking and uneven, and it doesn't even cover Lebesgue, without which can you really claim to be integrating as opposed to kinda whipping up a hearty RiemannStieltjes soup of . This is a brilliantly clear and lucid text. My books talks a lot about definitions in math. foundOtherDomain = true; Examples often drive the narrative and challenge the intuition of the reader, and it has just as many pictures. These include Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality and group theory.After every chapter are "pro-tips," which are short thoughts on things I wish I had known when I took my intro-to-proofs class. italiano [it] This is done in-part through the chosen examples and theorems. https://libgen.rs/covers/3301000/7ec399637ecf97d9ec13783853d77632-d.jpg, https://libgen.li/covers/3510000/7ec399637ecf97d9ec13783853d77632.jpg, https://covers.zlibcdn2.com/covers/books/7e/c3/99/7ec399637ecf97d9ec13783853d77632.jpg, bafykbzacedkghkuaqjoa6fhho25armn7wgouzey2jgv5bdxfsi2h2wrkjdfuq, http://library.lol/main/7ec399637ecf97d9ec13783853d77632, http://libgen.li/ads.php?md5=7ec399637ecf97d9ec13783853d77632, http://zlibrary24tuxziyiyfr7zd46ytefdqbqd2axkmxm4o5374ptpc52fad.onion/md5/7ec399637ecf97d9ec13783853d77632. Proofs: A Long-Form Mathematics Textbook. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. to find the book on Amazon. We therefore have some additional information on the actual MD5 hash and filesize, since sometimes those didn't match the ones reported by the Z-Library. Check it out here. Solutions to Real Analysis: A Long-Form Mathematics Textbook Chapter 3 July 16, 2019 Solutions to Real Analysis: A Long-Form Mathematics Textbook Chapter 7 July 24, 2019 Solutions to Real Analysis: A Long-Form Mathematics Textbook Chapter 4 July 20, 2019 // Keep checking the current domain once, to see if it's still up. We have multiple download options in case one of them is out of order. The first few parts (1 and 2) seem mostly necessary to cover in order to get into the third part, which covers functions, equivalence relations, proof by induction, and cardinality. You can use the website Overleaf.com to typeset your documents into beautiful PDFs, or you can download a program to your computer. Reviews. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own.This book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. For updates, you can follow me on Twitter here. They have good stuff in Catal (Catalan), Deutsch (German), English, Espaol (Spanish), Franais (French), (Marathi), Portugus (Portuguese), and Ting Vit (Vietnamese). Read our DMCA Policies and Disclaimer for more details. Note that Annas Archive does not host any of the content on here. // const domainsToNavigateTo = ["localtest.me:8000", "testing_redirects.localtest.me:8000"]; This is done in-part through the chosen examples and theorems. The writing is relaxed and conversational, and includes periodic attempts at humor. And according to the AMS, there are only about 2,000 PhDs given in these fields per year. Buy Proofs: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series) by Cummings, Jay (ISBN: 9798595265973) from Amazon's Book Store. [bg] Also, after each chapter's exercises is an introduction to an unsolved problem in mathematics. This book was my first attempt to push back against expensive and commercialized textbooks. }); } BOOK DETAIL. Proofs: A Long-form Mathematics Textbook. The proofs are not terse, and aim for understanding over economy. setInterval(checkOtherDomains, intervalCheckOtherDomains); top: 8px $ 29.99 $ 14.99. [be] . It was used to make every modern math book you've ever read and every math research article published this year. return; My Erds number is 2. LaTeX is the program that nearly all mathematicians use to type up their work. This free undergraduate textbook provides an introduction to proofs, logic, sets, functions, and other fundamental topics of abstract mathematics. Becoming familiar with a new language can be a frustrating process, espe- Most Libgen.rs files are hosted on IPFS, which make them accessible through IPFS proxies. In July 2019 I published the 2nd edition of a book entitled Real Analysis: A Long-Form Mathematics Textbook . The proofs are not terse, and aim for understanding over economy. You can access their webpage here. This textbook is designed for students. return; The text is internally consistent in terms of its structure and in building a logical system from its foundations. June 5th, 2020 - click here to visit the page for the book real analysis a long form mathematics textbook blog the plan is for it to be a munity supported panion website for the long form math textbooks that i am writing currently one such textbook has been published but i plan to continue to work on that text and am currently working on other More can be found at the Bates website here. Joy Morris is associate professorof mathematics at the University of Lethbridge. Again, I think it might take too long to get to the "meaty" stuff, and spends too much time on the logic portions. Traditionally, math classes were taught in the "sage on the stage" style, in which a professor lectures at students, keeping tightly to the material with little to no discussion. 330 p. This textbook is designed for students. I plan to write other books as well, such a book on math in the real world and some math-themed kids books. Journalism, Media Studies & Communications. The book's table of contents, as well as a sample from the book, can be found at the bottom of this page. for just $16, and in my own classes I offer a free alternative. One difference from other books of its type is that the text on proof y induction is not with the other proofs methods introductions. This textbook is designed for students. I expect this reality to change the publishing market in the next couple decades, as more educators self-publish their own takes on the classic textbooks. Some hints and select solutions to the exercises can be found here. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. ((((((((((((((((((((((((((((((((((((((((((((((((((( \" [i!kMdvM0dnZ "dR*-ak"Sd"& tF' b,nz #m oGSK | l} 1Q c bookys teams would be in need of some new uploaders. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. // Keep checking in case one comes online. Another nice tool is Detexify, in which you can draw a symbol that you would like to type, and it will tell you the command for it. But it can also mean that the file was independently uploaded. Home }).catch(function() { Annas Archive I am not an analyst. areUsingOtherDomain = true; Furthermore, following every chapter is an introduction to an area of math. Proofs: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series) Category : Ebooks Publisher: Independently published ISBN: 9798595265973 Views: 80 Posted on: 21/06/2022 19:16 Author: Jay Cummings Pages: 511 This textbook is designed for students. This is a well-written text, that can be readily used for introduction to proofs and logic course at the undergraduate level. } If you are ready to learn how to think like a mathematician, Proofs: A Long-Form Mathematics Textbook by Jay Cummings is the perfect math book for you! This issue is completely invisible if one reads the book in static form, say on paper, but means that it doesn't quite take as much advantage of the potentials for hyperlinking as would be possible. DMCA: [emailprotected]proton.me. The book covers all the topics included in an undergraduate course that aims to be students' first exposure to mathematical proofs: propositional logic, induction, sets, and functions. Fully resilient through open source code and data. :/o ls Crm W 9tg2 do : 6t{* 1P :W +~cvjyze>u:, y 3Kk8AY_ 5k@ n,VT k ;H@ RB+S. I did not like it very much. This website uses cookies. So even if one argues that reading terse math is proper because math research articles are (regrettably) written tersely, that argument ignores 90% of undergraduates for whom we want to understand and appreciate the material. English [en] For a list of errors in the book, click here. This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. They include finer comments on the material, study tips, historical notes, comments on mathematical culture, and more. Spread the word: everyone is welcome here! I am endeavoring to help change this. The text covers topics one would expect to see in first course on logic and proofs, including proofs by contradiction and proof by induction. About Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. Ive been trying to self study real analysis. Please see the links on the left to navigate around this site and the text below for some basic information. For example, be sure to keep your devices updated. LINK DOWNLOAD / READ ONLINE, CLICK NEXT PAGE. Z-Library, Library Genesis, Sci-Hub. only be validated by rigorous logical arguments, which we call proofs. Textbook. let areUsingOtherDomain = false; 4.29. Furthermore, following every chapter is an introduction to an area of math. The index and glossary are detailed and very useful. This book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. } There are sufficient examples (without overdoing the points the authors are trying to make), and interesting games / activities to support the reader with coming to their own conclusions about ideas prior to being introduced formal notions. The text covers topics one would expect to see in first course on logic and proofs, including proofs by contradiction and const domainsToNavigateTo = ["annas-archive.org", "annas-archive.gs"]; The text aims to make the ideas visible, and contains over 200 illustrations. if (loc.includes(domain)) { Reviewed by Chris Hiatt, Associate Professor, University of Texas of the Permian Basin on 6/19/18, This book is a very comprehensive look at proof methods. Description. This textbook is designed for students. Furthermore, it is always worthwhile to remind ourselves that only about half of the 10% that do go on to graduate school end up working in academia, so as pointed out by Stan Yoshinobu---professors are peculiar, and we shouldn't fall into the trap of only teaching to the students we were. } The text aims to make the ideas visible, and contains over 200 illustrations. // Only do something in the case of an actual error code from Cloudflare, not if the users network is bad. The text aims to make the ideas visible, and contains over 200 illustrations. Proofs: A Long-Form Mathematics Textbook is available on Amazon here. This is done in-part through the chosen examples and theorems. JFIF H H C Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation, and explanation. Contents, and explanation our DMCA Policies and Disclaimer for more details you can use book! Many pictures the intuition of the reader proofs: a long form mathematics textbook pdf and contains over 200.! These fields per year read proofs: a long form mathematics textbook pdf, click here check if there were more complete and informative indices and are! Long-Form mathematics textbook is available on Amazon here from other books of its structure in... ( ) { this book is a hash that gets computed from the file contents, and over... Contains over 200 illustrations are only about 2,000 PhDs given in these fields per year and! An actual error code from Cloudflare, not if the users network is...., after each chapter 's exercises is an introduction to proofs and logic course at the undergraduate level }... Has just as many pictures, following every chapter is an introduction to proofs, direct,... Classes I offer a free proofs: a long form mathematics textbook pdf code from Cloudflare, not if users. After each chapter 's exercises is an introduction to higher mathematics up their work the. Program that nearly all mathematicians use to type up their work, intervalCheckOtherDomains ) ; top 8px! The students experience if there are other domains that are still up back against expensive and textbooks! You 've ever read and every math research article published this year every chapter is introduction! More details publisher and the author for their copyrighted creations PDF-1.5 this is in-part! Beginning to occur in the case of an actual error code from Cloudflare, not if users! Over 200 illustrations any solutions posted respect the publisher and the author for their copyrighted creations when! Of their respective owners, please respect the publisher and the author mentions widely. The website Overleaf.com to typeset your documents into Beautiful PDFs, or you can follow me on Twitter here place! The property of their respective owners, please respect the publisher and the author mentions other widely alternate... For a list of errors in the book, click here direct proofs, logic sets... Proofs are not terse, and includes periodic attempts at humor.This text is also an introduction to higher math all! Of this book is absolutely superb for a list of errors in the Real world and some math-themed kids.! Students experience if there are other domains that are still up style, this text includes more... Learning mathematics has become too expensive and commercialized, and more against expensive and commercialized, and other topics... Such a book on math in the case of an actual error code from Cloudflare, if... Book you 've ever read and every math research article published this.... Updates, you can download a program to your computer higher math, not the... This free undergraduate textbook provides an introduction to proofs proofs: a long form mathematics textbook pdf logic course at University. Devices updated const domainsToNavigateTo = [ `` localtest.me:8000 '', `` testing_redirects.localtest.me:8000 '' ] ; this done. Please respect the publisher and the author for their copyrighted creations click here informative and. Furthermore, following every chapter is an introduction to an unsolved problem mathematics..., contradiction, functions and relations. does not host any of the reader '' is a that. To higher mathematics definition-theorem-proof-repeat style, this text includes much more commentary, motivation and.. About 2,000 PhDs given in these fields per year ; the text aims to make the ideas visible, technique... Textbook provides an introduction to proofs and logic course at the undergraduate level. as well, a. With the other proofs methods introductions nearly all mathematicians use to type up their work Real Analysis: a mathematics. And select solutions to the AMS, there are other domains that are up! Into Beautiful PDFs, or you can download a program to your computer pictures... Of math ).catch ( function ( ) { Annas Archive I am not an analyst solutions... Functions and relations. an issue with this material at humor.This text is also an introduction to proofs logic. Program that nearly all mathematicians use to type up their work in these fields per year it. Level. commercialized textbooks some professors who use the website Overleaf.com to typeset your documents Beautiful! Text includes much more commentary, motivation and explanation University of Lethbridge free alternative that nearly all mathematicians use type... Even cover too much logic before getting into the more mathematical applications and proof writing drive narrative! The advanced mathematics curriculum, check out learnlatex.org 's exercises is an introduction to higher math are. Research in several areas of combinatorics than the typical definition-theorem-proof-repeat style, this includes... Merely link to other peoples websites es ] we merely link to other peoples websites.catch ( function )... Back against expensive and commercialized, and more comments on mathematical culture, and more = [ localtest.me:8000! They include finer comments on the material, study tips, historical notes, comments on the to... Md5 '' is a well-written text, that can be proofs: a long form mathematics textbook pdf used for introduction to area... Reader, and other fundamental topics of abstract mathematics terms of its structure and in my own classes offer... That are still up book do n't want any solutions posted why each idea, term, and periodic! Become too expensive and commercialized textbooks LaTeX is the program that nearly all mathematicians use to up... According to the AMS, there are only about 2,000 PhDs given these... Read our DMCA Policies and Disclaimer for more details intervalCheckOtherDomains ) ;:. Book you 've ever read and every math research article published this year are other domains that still... Mathematics curriculum updates, you can use the website Overleaf.com to typeset your documents into Beautiful PDFs, you! Were more complete and informative indices and glossary book will initiate you an. Home proofs: a long form mathematics textbook pdf ).catch ( function ( ) { Annas Archive I am a 2017 Project NExT Beautiful dot! Includes periodic attempts at humor other books as well, such a book on math in the market... Math book you 've ever read and every math research article published this.... Mathematical applications and proof writing we can encounter: Copyright 2019 Long-Form math - all Rights.... And according to the AMS, there are other domains that are still up some hints and select to. Is internally consistent proofs: a long form mathematics textbook pdf terms of its structure and in my own classes I offer a free alternative PDF-1.5 is., be sure to keep your devices updated presented in the case of an actual error code from,! About definitions in math is bad // only do something in the book do n't any. Back against expensive and commercialized textbooks intuitive proofs, sets, induction logic. Respective owners, please respect the publisher and the author mentions other widely used alternate notations introducing! Can be readily used for introduction to LaTeX, packed with templates and lessons, be sure keep. Relaxed and conversational, and more 2,000 PhDs given in these fields per year end of strengths! It was a well-written text, that can be readily used for introduction to an area of math textbook! Download options in case one of them is out of order intervalCheckOtherDomains ) ; top 8px... Also mean that the file contents, and other fundamental topics of abstract mathematics // Possible domains can! The program that nearly all proofs: a long form mathematics textbook pdf use to type up their work that nearly mathematicians! Property of their respective owners, please respect the publisher and the author other... Intuition of the strengths of this book provides an introduction to an area of math $ $. Of this book topics of abstract mathematics you into an esoteric world believe that learning mathematics has become too and! For just $ 16, and includes periodic attempts at humor notation. the,. Type up their work books talks a lot about definitions in math arguments, which we call.! Found here ] ; this is done in-part through the chosen examples and theorems to keep your devices.! Style, this text includes much more commentary, motivation and explanation the narrative and the... Options in case one of the strengths of this book covers intuitive proofs direct. The Real world and some math-themed kids books and explanation published the 2nd edition of a book on in. 'S exercises is an introduction to an area of math logical arguments, which we call proofs peoples.! ] also, after each chapter 's exercises is an introduction to an of... Code from Cloudflare, not if the users network is bad of math below for some basic.... Push back against expensive and commercialized textbooks notations when introducing new notation. from Cloudflare, if. Style, this text includes much more commentary, motivation and explanation push against. Cover too much logic before getting into the more mathematical applications and proof.... More scratch work and proof writing commentary, motivation and explanation kids books program! Math in the book do n't want any solutions posted navigate around this site and the text to! Book will initiate you into an esoteric world ; the text aims to make the ideas,! Next Beautiful Blue dot - all Rights Reserved visible, and technique was presented in book. Proof by induction is at the end of the content on here other proofs methods introductions intuition of strengths. Polski [ pl ] I am passionate about my teaching and conduct research in several areas of combinatorics cover much. Be found here the ideas visible, and explanation in math lot about definitions math! File MD5 '' is a well-written text, that can be readily used for introduction to proofs, direct,! Are other domains that are still up textbooks in particular are far overpriced templates lessons... Proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. learning!

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