Matematikkens fundament

Euklids ELEMENTER er et hovedværk ikke alene i matematikken, men i hele den vestlige civilisation. Det er det ældste aksiomatisk opbyggede værk fra 300 f.Kr. med linjer tilbage til Platons og Aristoteles’ filosofi og fremad mod Archimedes og den senere matematik.Dette er den første danske oversættelse af Euklids plangeometri samlet i hans ELEMENTER I-VI siden Thyra Eibes oversættelse fra 1897-1912.Den nye oversættelse er udsprunget af et ønske om at vise Euklids værk i lyset af den moderne forståelse af græsk matematik som en udløber af samtidens filosofiske retninger; Platon og Aristoteles citeres derfor flittigt i den lange indledning. På samme måde er bogen udstyret med en redegørelse for Euklids sprog og dets mangel på symboler af enhver art - i modsætning til den hidtidige oversættelse, der i overensstemmelse med praksis i det sene 19. århundrede gør brug af symboler fra samtidens matematiske sprog.Desuden er der i appendices vigtige matematikhistoriske tekster og emner (Proklos' oversigt over geometrier, Hilberts aksiomer, Hippias' Quadratrix) og andre hjælpemidler, som ikke findes andetsteds.Bogen henvender sig til en bred gruppe læsere: professionelle matematikere fx i gymnasiet og på universitetet, hvor der undervises i matematikkens historie og videnskabsteori, filosofi- og antikstuderende sammesteds og endelig den ret store gruppe mennesker, der interesserer sig for antik filosofi og matematik.Euklids ELEMENTER består af 13 mindre bøger. Gyldendal samler dem i fire bind. Dette første bind indeholder Bog I-VI.  Euklids ELEMENTER I-VI er indledt, oversat og forsynet med bemærkninger af Claus Glunk, Hanne Eggert Strand, Chr. Marinus Taisbak og Chr. Gorm Tortzen.

Den græske matematiker Euklids lærebog ELEMENTER er, hvis man ser bort fra Bibelen, den mest trykte lærebog gennem tiderne. ELEMENTER er skrevet omkring 300 f. Kristus, og er en samling af i alt 13 bøger, der præsenterer den samlede viden om matematik og geometri i tiden.ELEMENTER er et helt uomgængeligt værk for den matematik- og filosofiinteresserede.Bøgerne 1-6, der behandler plangeometrien, er udgivet i foreløbig to oplag, og nu udgives bøgerne 7-9, der omhandler naturlige tal, som bind 2 i det samlede værk.

'An invaluable companion for anyone who wants a deep understanding of what's under the hood of often inscrutable machines' Melanie Mitchell A rich, narrative explanation of the mathematics that has brought us machine learning and the ongoing explosion of artificial intelligenceMachine-learning systems are making life-altering decisions for us: approving mortgage loans, determining whether a tumour is cancerous, or deciding whether someone gets bail. They now influence discoveries in chemistry, biology and physics - the study of genomes, extra-solar planets, even the intricacies of quantum systems.We are living through a revolution in artificial intelligence that is not slowing down. This major shift is based on simple mathematics, some of which goes back centuries: linear algebra and calculus, the stuff of eighteenth-century mathematics. Indeed by the mid-1850s, a lot of the groundwork was all done. It took the development of computer science and the kindling of 1990s computer chips designed for video games to ignite the explosion of AI that we see all around us today. In this enlightening book, Anil Ananthaswamy explains the fundamental maths behind AI, which suggests that the basics of natural and artificial intelligence might follow the same mathematical rules.As Ananthaswamy resonantly concludes, to make the most of our most wondrous technologies we need to understand their profound limitations - the clues lie in the maths that makes AI possible.

Hej logik giver en pædagogisk indføring i klassisk logik. Med eksempler og øvelser guider forfatteren læseren igennem centrale temaer i doms- og prædikatslogikken: Hvordan gyldighed hænger sammen med logisk form, hvordan man afdækker logiske strukturer ved at formalisere elementerne i et argument, og hvordan man analyserer logisk form ved at bruge sandhedstabeller og semantiske træer. Bogen kan bruges som lærebog til logikkurser i gymnasiet og på universitetet. Samtidig gør en filosofisk perspektivering af grundtemaerne den også interessant for den mere logikkyndige.Titlen henviser til 'Hej logikmaskinen', der er bogens nye måde at forstå logikkens centrale funktionsbegreb på. Mange finder funktionsbegrebet abstrakt og utilgængeligt, men i Hej logik bliver det forståeligt og intuitivt.

Bind 3 i den nye oversættelse af den græske matematiker Euklids lærebog ELEMENTER omhandler kommensurabilitet. ELEMENTER er skrevet omkring 300 f. Kristus og er en samling af i alt 13 bøger, der præsenterer den samlede viden om matematik og geometri i tiden.Engang omkring år 500 f.Kr. opdagede nogle pythagoræiske matematikere et matematisk problem, som de meget tøvende gik i gang med at finde løsninger på: Nogle tal og linjestykker kan måles med den samme målestok, andre kan ikke.  Et kendt eksempel er et kvadrat på 2x2 fod (eller meter), hvis diagonal målenheden ikke ’går op’ i.Linjer er med andre ord kommensurable eller inkommensurable med hinanden. Men hvordan hænger det sammen?I de foregående ni bøger i Elementerne har Euklid fremstillet geometrien i sin klassiske form på en ny og overskuelig måde, der gjorde alle tidligere geometrier forældede, men i Bog X tager han fat på læsningerne i en helt ny og uudforsket gren af matematikken, læren om kommensurabilitet, som havde fået et afgørende spring fremad med matematikerne Theodoros og Theaitetos i begyndelsen af 300-tallet f.Kr., men endnu ikke var blevet lærebogsstof. Bog X, som i omfang udgør næsten en fjerdedel af hele Euklids Elementer, bærer som frontforskning præg af at skulle fremstille en ny og uprøvet videnskabsgren i et forsøg på at bringe orden og system i den nye erkendelse.Euklids ELEMENTER er et hovedværk ikke alene i matematikken, men i hele den vestlige civilisation og er et helt uomgængeligt værk for den matematik- og filosofiinteresserede.Bogen henvender sig til en bred gruppe læsere: professionelle matematikere fx i gymnasiet og på universitetet, hvor der undervises i matematikkens historie og videnskabsteori, filosofi- og antikstuderende sammesteds og endelig den ret store gruppe mennesker, der interesserer sig for antik filosofi og matematik.Euklids ELEMENTER bog X er indledt, oversat og forsynet med bemærkninger af Claus Glunk, Hanne Eggert Strand, Chr. Marinus Taisbak og Chr. Gorm Tortzen.

The Journal of Applied Logics - IfCoLog Journal of Logics and their Applications (FLAP) covers all areas of pure and applied logic, broadly construed. All papers published are free open access, and available via the College Publications website. This Journal is open access, puts no limit on the number of pages of any article, puts no limit on the number of papers in an issue and puts no limit on the number of issues per year. We insist only on a very high academic standard, and will publish issues as they come.

Algebra con elementi di logica matematica e teoria degli insiemi.

This book is an introduction to the logic of partitions on a set as well as the (quantum) logic of partitions (direct-sum decompositions or DSDs) on a vector space. Partitions of a set are categorically dual to subsets of a set. Thus the logic of partitions is, in that sense, the dual to the Boolean logic of subsets (usually presented as the special case of propositional logic).Since partitions can be seen as the inverse image partitions of random variables or numerical attributes, partition logic is the logic of random variables or numerical attributes (abstracted from the actual values). On the lattice of partitions of an arbitrary unstructured set, there is a rich algebraic structure of dual operations of implication and co-implication - resembling a non-distributive version of Heyting and co-Heyting algebras.

Mathematik für die Informatik: Dieses Buch bringt Ihnen die Grundlagen beiDieses Lehrbuch bietet eine Einführung in die grundlegenden Begriffe und Strukturen der Mathematik, welche am Anfang eines Informatikstudiums relevant sind. Weiterhin demonstriert es Anwendungen von mathematischen Konzepten und Methoden in der Informatik. Diese betreffen insbesondere formale Methoden der Programmverifikation und -entwicklung und den Entwurf von generischen Programmen. Ein spezielles Konzept mit einer leicht verständlichen Vermittlung des Stoffs, vielen Beispielen mit Rückgriffen auf die Schulmathematik und detaillierten Beweisen (verbunden mit der Erklärung des logischen Hintergrunds) erleichtert den Einstieg in die Mathematik an einer wissenschaftlichen Hochschule. Dadurch werden die Studierenden auch auf spätere Begriffe und tiefergehende Anwendungen der Mathematik in der Informatik gut vorbereitet. Die 5. Auflage erweitert den Inhalt um die Grundlagen der linearen Algebra. Die Übungsaufgaben zu den einzelnen Kapitel helfen, das Erlernte zu festigen und zu kontrollieren. Zahlreiche Lösungsvorschläge am Ende des Buchs ermöglichen die Überprüfung der eigenen Lösungen.Der InhaltMengentheoretische Grundlagen ¿ Logische Grundlagen ¿ Allgemeine direkte Produkte und Datenstrukturen ¿ Mathematische Beweise ¿ Spezifikation und Programmverifikation ¿ Spezielle Funktionen ¿ Spezielle Relationen und gerichtete Graphen ¿ Elementare Kombinatorik und ungerichtete Graphen ¿ Diskrete Wahrscheinlichkeitstheorie ¿ Generische Programmierung ¿ Grundbegriffe algebraischer Strukturen ¿ Formale Einführung der natürlichen Zahlen - Grundbegriffe der linearen AlgebraMit diesem Buch gelingt der Einstieg ins Informatik-StudiumMit diesem Buch schaffen Sie eine solide Basis für die Mathematikausbildung im Rahmen desInformatikstudiums. Zudem sind Sie durch die vorgestellten Problemstellungen in der Lage, selbstständig mathematische Konzepte und Methoden anzuwenden. Zielgruppen dieses Buchs über Mathematik in der Informatik sind Bachelor-Studierende in den ersten Studiensemestern folgender Fachbereiche:InformatikMathematikIngenieurwissenschaften

This volume of the Logica Yearbook series brings together articles presented at the annual international symposium Logica 2022, Teplá, the Czech Republic. The articles range over mathematical and philosophical logic, history and philosophy of logic, and the analysis of natural language.

This book constitutes the revised selected papers of the 5th International Workshop on Dynamic Logic. New Trends and Applications, DaLí 2023, held in Tbilisi, Georgia, during September 15¿16, 2023. The 8 full papers in this book were carefully reviewed and selected from 10 submissions. They deal with new trends and applications in the area of Dynamic Logic.

Fuzzy logic systems have been a hot topic in the scientific and academic community for more than half a century. The idea of making machines behave and make decisions like humans do is astounding. The development and implementation of fuzzy logic systems can be seen in various real physical applications in daily human life. The methods employed using fuzzy logic have resulted in innovative technologies. This book provides insights into understanding the principles and concepts behind the advances of fuzzy logic systems. It presents ideas concerning fuzzy logic systems and their technological applications. The book is arranged into two sections on theories and foundations of fuzzy logic systems and implementations of fuzzy logic systems in service to the community.

The Journal of Applied Logics - IfCoLog Journal of Logics and their Applications (FLAP) covers all areas of pure and applied logic, broadly construed. All papers published are free open access, and available via the College Publications website. This Journal is open access, puts no limit on the number of pages of any article, puts no limit on the number of papers in an issue and puts no limit on the number of issues per year. We insist only on a very high academic standard, and will publish issues as they come.

Lógica Dialógica. Reglas y ejercicios para hacer lógica con diálogos constituye la primera introducción a la Lógica dialógica en castellano dirigida a la práctica de la dialógica y que contiene comentarios precisos sobre ejercicios y soluciones tanto de lógica proposicional como de lógica clásica de primer orden, intuicionista y elementos de lógica modal proposicional. Es la primera parte de la obra Los Diálogos de la Lógica, concebida en dos textos autónomos separados sobre lógica dialógica. Los dos textos deberían ofrecer conjuntamente una visión técnica y filosófica completa del enfoque dialógico de la lógica

Math Mammoth Grade 8 Answer Keys includes answer keys to: Math Mammoth Grade 8-A worktext;Math Mammoth Grade 8-B worktext;End-of-chapter tests;End-of-year test;Cumulative reviews.

Math Mammoth Grade 8-A Worktext is the first student book for eighth grade mathematics, and part of the Math Mammoth Grade 8 complete curriculum.This book starts with a study of exponent laws, using both numerical and algebraic expressions. The first chapter also covers scientific notation (both with large and small numbers), significant digits, and calculations with numbers given in scientific notations. In chapter 2, students learn about geometric transformations (translations, reflections, rotations, dilations), common angle relationships, and volume of prisms, cylinders, spheres, and cones. Next, in chapter 3, our focus is on linear equations. Students both review and learn more about solving linear equations, including equations whose solutions require the usage of the distributive property and equations where the variable is on both sides. Chapter 4 presents an introduction to functions. Students construct functions to model linear relationships, learn to use the rate of change and initial value of the function, and they describe functions qualitatively based on their graphs. Features Math Mammoth focuses on conceptual understanding. It explains the "WHY", so students can understand the math, not just learn "HOW" to do it.The curriculum is mastery-oriented. This means it concentrates fairly long on a topic, delving into its various aspects. This promotes conceptual understanding, as opposed to spiral curricula that often tend to jump from topic to topic too much.It requires very little teacher preparation. :)The curriculum has no separate teacher's manual nor is it scripted. The introduction to each chapter has some notes for the teacher concerning the material in the chapter. All the instruction is written directly to the student in the worktext.After each chapter introduction, you will find a list of Internet links and resources (games, quizzes, animations, etc.) that can be used for fun, illustrations, and further practice.

Math Mammoth Grade 8-B Worktext is the second student book for eighth grade mathematics, and part of the Math Mammoth Grade 8 complete curriculum. The book starts with chapter 5, graphing linear equations. Students study slope, and graph and write linear equations in slope-intercept and standard forms. They also learn about parallel and perpendicular lines.In chapter 6, students delve into square roots, cube roots, and irrational numbers and solve equations involving square and cube roots. The rest of the chapter is spent studying the Pythagorean Theorem and its applications. Next, in chapter 7, we focus on systems of linear equations. Students learn the three basic techniques for solving these and also solve a variety of problems that are solved with a pair of linear equations. The last chapter covers statistical topics: scatter plots and two-way tables.Features Math Mammoth focuses on conceptual understanding. It explains the "WHY", so students can understand the math, not just learn "HOW" to do it.The curriculum is mastery-oriented. This means it concentrates fairly long on a topic, delving into its various aspects. This promotes conceptual understanding, as opposed to spiral curricula that often tend to jump from topic to topic too much.It requires very little teacher preparation. :) The curriculum has no separate teacher's manual nor is it scripted. The introduction to each chapter has some notes for the teacher concerning the material in the chapter. All the instruction is written directly to the student in the worktext.Each chapter introduction also includes a list of Internet links and resources (games, quizzes, animations, etc.) that can be used for fun, illustrations, and further practice.

Math Mammoth Grade 8 Tests & Cumulative Reviews contains the end-of-chapter tests, the comprehensive end-of-year test, and optional cumulative review lessons for Math Mammoth grade 8 complete curriculum.

Taking an amusing and digestible look at the usually dry world of probability and statistics, this is the ultimate guide to how you can incorporate them into everyday life, from one of the world's most sought-after experts in game theory. This is the only book you need to become a statistics whizz! Numbers are everywhere – food packaging, weather forecasts, social media, adverts, and more. You can’t escape them. But you can learn to understand them – and avoid being fooled! This book breaks down the key fundamentals in statistics in a fun and accessible way so that you can understand the numbers that occupy your life. • Make sense of sports stats – discover who is the greatest scorer of all time • Learn to interpret scientific studies and how they’re reported in the media so you’re never misled again • Discover tips and tricks to make you a more successful gambler • Explore what role stats has to play in flat-earth conspiracy arguments • Read about misunderstood probabilities in the Sally Clarke and OJ Simpson trials With easy-to-follow explanations, tables, graphs, and real-life examples, this book helps you evaluate your options, calculate your chances of success, and make better decisions.

Math Mammoth Grade 6 Tests & Cumulative Reviews (Canadian version) contains the end-of-chapter tests, the end-of-year test, and cumulative reviews lessons to match Math Mammoth Grade 6 curriculum (Canadian version).

Math Mammoth Grade 6-A worktext (Canadian version) is the first student book for sixth grade, and part of the Math Mammoth Grade 6 complete curriculum. In this book, students encounter the beginnings of algebra, learning about algebraic expressions, one-variable equations and inequalities, integers, and ratios. We also review and deepen the students' understanding of decimals, and study ratios and percent as new topics for 6th grade. The year starts out with a review of the four operations with whole numbers (including long division), place value, and rounding. Students are also introduced to exponents and do some problem solving.Chapter 2 starts the study of algebra topics, delving first into expressions and equations. Students practice writing expressions in many different ways, and use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations. We also study briefly inequalities and using two variables. Chapter 3 has to do with decimals. This is a long chapter, as we review all of decimal arithmetic, just using more decimal digits than in 5th grade. Students also convert measuring units in this chapter. Ratios is a new topic (chapter 4). Students are already familiar with finding fractional parts from earlier grades, and now it is time to advance that knowledge into the study of ratios, which arise naturally from dividing a quantity into many equal parts. We study such topics as rates, unit rates, equivalent ratios, and problem solving using bar models.Percent (chapter 5) is an important topic to understand thoroughly, because of its many applications in real life. The goal of this chapter is to develop a basic understanding of percent, to see percentages as decimals, and to learn to calculate discounts. This curriculum is essentially the same as the U.S. version of Math Mammoth, only customized for the Canadian audience in these aspects:The curriculum uses the metric measurement units. Since Canadians do use imperial units, such as inches and pounds, in some contexts, the curriculum does mention them, but not often.The spelling conforms mostly to American English, taking into account a few key differences where Canadian English follows British English.Numbers are formatted with a space as a thousands separator, and a point as a decimal separator.Page (paper) size is Letter.The US version of the curriculum is aligned to the Common Core Standards, which means this curriculum may not be properly aligned to the sixth grade standards in any Canadian province. However, you can probably find material for any missing topics in the neighbouring grade levels of Math Mammoth materials.Available separately, Math Mammoth Grade 6-B worktext covers the second half of 6th grade math. Also available are the answer keys and a book containing cumulative reviews and chapter tests.

This book includes answers for the Math Mammoth Grade 6-A and 6-B student worktexts (Canadian version), for all the chapter tests and end-of-year test, and for the cumulative reviews for grade 6. The other parts of the 6th grade curriculum (the 6-A and 6-B worktexts and the cumulative reviews & tests book) may be purchased separately.