CAPES / Este trabalho tem como objetivo aprofundar o conhecimento dos professores do ensino médio fundamental a respeito de congruência e polinômios. Apesar de congruência não ser abordado nas escolas, este assunto justifica alguns conceitos repassados aos alunos, como por exemplo a divisibilidade de um número por outro. A congruência ainda pode auxiliar na verificação de raízes de polinômios. Aqui, os polinômios são tratados como elementos de um anel, o anel dos polinômios, e vários resultados utilizados em sala de aula são justificados a partir da estrutura desse anel. Com esses dois conceitos, ainda e feito um breve estudo de congruência polinomial. / The aim of this work is to deepen the knowledge of elementary and high school teachers about congruence and polynomials. Although congruence is not studied in schools, this subject justifies some concepts passed to the students, such as the divisibility of one number by another. The congruence can also help to verify roots of polynomials. Here, polynomials are treated as elements of a ring, the ring of polynomials, and several results used in the classroom are justified from the structure of this ring. These concepts are used for a brief study of polynomial congruence.

Orientador: Prof. Dr. Eduardo Guéron / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, Santo André, 2018. / Calculadoras são aparelhos comuns no cotidiano do homem moderno, contudo, os conceitos matemáticos envolvidos em sua concepção ainda são conhecidos por poucos. Durante séculos, a obstinação da humanidade em construir máquinas capazes de computar de forma autônoma resultou tanto no surgimento dos atuais computadores, como também em um magnífico legado de conhecimentos matemáticos agregados a tal conquista. Conteúdos tais como congruências e álgebra booleana suscitaram a revolução dos sistemas informatizados e tem sido amplamente explorados por meio de inúmeras aplicações, nossa trajetória perpassou pela aritmética modular, o teorema de Euler-Fermat e as classes residuais, além de bases numéricas, tópicos de eletrônica digital e funções booleanas, com foco no desenvolvimento de circuitos lógicos e o engendrar de componentes eletrônicos, que configuram a base para idealização e construção de calculadoras que efetuem as operações aritméticas em bases arbitrárias, objetivo preponderante deste trabalho. O esmiuçar das etapas de construção das calculadoras, viabiliza o aprofundamento dos conceitos matemáticos que a fomentaram. A abordagem dos temas supracitados culmina para aprimorar e evidenciar a aplicabilidade da matemática à essência da era moderna. / Calculators are common apparatuses in the everyday of modern man, however, the mathematical concepts involved in its conception are still known by few. For centuries, mankind¿s obstinacy in building machines capable of computing autonomously resulted in both the emergence of current computers and a magnificent legacy of mathematical knowledge added to such achievement. Contents such as congruences and Boolean algebra have aroused the revolution of computerized systems and it has been extensively explored through numerous applications, our trajectory ran through modular arithmetic, Euler-Fermat¿s theorem and residual classes, as well as numerical bases, topics of digital electronics and Boolean functions, focusing on the development of logic circuits and the generation of electronic components, which form the basis for the design and construction of calculators that perform arithmetic operations on arbitrary bases, a preponderant objective of this work. The to detail of the construction steps of the calculators, enables the deepening of the mathematical concepts that fomented it. The approach to the aforementioned themes culminates in improving and evidencing the applicability of mathematics to the essence of the modern era.

<p>The research work carried out in this thesis is aimed&nbsp;&nbsp; at fuzzifying algebraic and relational structures in the framework of Ω-sets, where Ω is a complete lattice.<br />Therefore we attempt to synthesis universal algebra and fuzzy set theory. Our&nbsp; investigations of Ω-algebraic structures are based on Ω-valued equality, satisability of identities and cut techniques. We introduce Ω-algebras, Ω-valued congruences,&nbsp; corresponding quotient&nbsp; Ω-valued-algebras and&nbsp; Ω-valued homomorphisms and we investigate connections among these notions. We prove that there is an Ω-valued homomorphism from an Ω-algebra to the corresponding quotient Ω-algebra. The kernel<br />of an Ω-valued homomorphism is an Ω-valued congruence. When dealing with cut structures, we prove that an Ω-valued homomorphism determines classical homomorphisms among the corresponding quotient structures over cut&nbsp; subalgebras. In addition, an&nbsp; Ω-valued congruence determines a closure system of classical congruences on cut subalgebras. In addition, identities are preserved under Ω-valued homomorphisms. Therefore in the framework of Ω-sets we were able to introduce Ω-attice both as an ordered and algebraic structures. By this Ω-poset is defined as an Ω-set equipped with&nbsp; Ω-valued order which is&nbsp; antisymmetric with respect to the corresponding Ω-valued equality. Thus defining the notion of pseudo-infimum and pseudo-supremum we obtained the definition of Ω-lattice as an ordered structure. It is also defined that the an Ω-lattice as an algebra is a bi-groupoid equipped with an Ω-valued equality fulfilling some particular lattice Ω-theoretical formulas. Thus using axiom of choice we proved that the two approaches are equivalent. Then we also introduced the notion of complete Ω-lattice based on Ω-lattice. It was defined as a generalization of the classical complete lattice.<br />We proved results that characterizes Ω-structures and many other interesting results.<br />Also the connection between Ω-algebra and the notion of weak congruences is presented.<br />We conclude with what we feel are most interesting areas for future work.</p> / <p>Tema ovog rada je fazifikovanje algebarskih i relacijskih struktura u okviru omega- skupova, gdeje Ω kompletna mreza. U radu se bavimo sintezom oblasti univerzalne algebre i teorije rasplinutih (fazi) skupova. Na&scaron;a istraživanja omega-algebarskih struktura bazirana su na omega-vrednosnoj jednakosti,zadovoljivosti identiteta i tehnici rada sa nivoima. U radu uvodimo omega-algebre,omega-vrednosne kongruencije, odgovarajuće omega-strukture, i omega-vrednosne homomorfizme i istražujemo veze izmedju ovih pojmova. Dokazujemo da postoji Ω -vrednosni homomorfizam iz Ω -algebre na odgovarajuću količničku Ω -algebru. Jezgro Ω -vrednosnog homomorfizma je Ω- vrednosna kongruencija. U vezi sa nivoima struktura, dokazujemo da Ω -vrednosni homomorfizam odredjuje klasične homomorfizme na odgovarajućim količničkim strukturama preko nivoa podalgebri. Osim toga, Ω-vrednosna kongruencija odredjuje sistem zatvaranja klasične kongruencije na nivo podalgebrama. Dalje, identiteti su očuvani u Ω- vrednosnim homomorfnim slikama.U nastavku smo u okviru Ω-skupova uveli Ω-mreže kao uredjene skupove i kao algebre i dokazali ekvivalenciju ovih pojmova. Ω-poset je definisan kao Ω -relacija koja je antisimetrična i tranzitivna u odnosu na odgovarajuću Ω-vrednosnu jednakost. Definisani su pojmovi pseudo-infimuma i pseudo-supremuma i tako smo dobili definiciju Ω-mreže kao uredjene strukture. Takodje je definisana Ω-mreža kao algebra, u ovim kontekstu nosač te strukture je bi-grupoid koji je saglasan sa Ω-vrednosnom jednako&scaron;ću i ispunjava neke mrežno-teorijske formule. Koristeći aksiom izbora dokazali smo da su dva pristupa ekvivalentna. Dalje smo uveli i pojam potpune Ω-mreže kao uop&scaron;tenje klasične potpune mreže. Dokazali smo jo&scaron; neke rezultate koji karakteri&scaron;u Ω-strukture.Data je i veza izmedju Ω-algebre i pojma slabih kongruencija.Na kraju je dat prikaz pravaca daljih istrazivanja.</p>

<p>In this work a comparative analysis of several approaches to fuzzy algebraic structures and comparison of previous approaches to the recent one developed at University of&nbsp; Novi Sad has been done. Special attention is paid to reducts and expansions of algebraic structures in fuzzy settings. Besides mentioning all the relevant algebras and properties developed in this setting, particular new algebras and properties are developed and investigated. Some new structures, in particular Omega Boolean algebras, Omega Boolean lattices and Omega Boolean rings are developed in the framework of omega structures. Equivalences among these structures are elaborated in details. Transfers from Omega groupoids to Omega groups and back are demonstrated. Moreover, normal subgroups are introduced in a particular way. Their connections to congruences are elaborated in this settings. Subgroups, congruences and normal subgroups are investigated for Ω-groups. These are latticevalued algebraic structures, defined on crisp algebras which are not necessarily groups, and in which the classical equality is replaced by a lattice-valued one. A normal Ω-subgroup is defined as a particular class in an Ω-congruence. Our main result is that the quotient groups over cuts of a normal Ω- subgroup of an Ω-group G, are classical normal subgroups of the corresponding quotient groups over G. We also describe the minimal normal Ω-subgroup of an Ω-group, and some other constructions related to Ω-valued congruences.Further results that are obtained are theorems that connect various approaches of fuzzy algebraic structures. A special notion of a generalized lattice valued Boolean algebra is introduced. The universe of this structure is an algebra with two binary, an unary and two nullary operations (as usual), but which is not a crisp Boolean algebra in general. A main element in our approach is a fuzzy&nbsp; quivalence relation such that the Boolean algebras identities are approximately satisfied related to the considered fuzzy equivalence. Main properties of the new introduced notions are proved, and a connection with the notion of a structure of a generalized fuzzy lattice is provided.</p> / <p>Ovaj rad bavi se komparativnom analizom različitih pristupa rasplinutim (fazi) algebarskim strukturama i odnosom tih struktura sa odgovarajućim klasičnim&nbsp;&nbsp; algebrama. Posebna pažnja posvećena je poredenju postojećih pristupa ovom&nbsp;&nbsp; problemu sa novim tehnikama i pojmovima nedavno razvijenim na Univerzitetu u Novom Sadu. U okviru ove analize, proučavana su i pro&scaron;irenja kao i redukti algebarskih struktura u kontekstu rasplinutih algebri. Brojne važne konkretne algebarske strukture istraživane su u ovom kontekstu, a neke nove uvedene su i ispitane. Bavili smo se detaljnim istrazivanjima Ω-grupa, sa stanovista kongruencija, normalnih podgrupa i veze sa klasicnim grupama. Nove strukture koje su u radu uvedene u posebnom delu, istrazene su sa aspekta svojstava i medusobne ekvivalentnosti. To su Ω-Bulove algebre, kao i odgo-varajuce mreže i Bulovi prsteni. Uspostavljena je uzajamna ekvivalentnost tih struktura analogno odnosima u klasičnoj algebri. U osnovi na&scaron;e konstrukcije su mrežno vrednosne algebarske strukture denisane na klasičnim algebrama koje ne zadovoljavaju nužno identitete ispunjene na odgovarajucim klasičnim strukturama (Bulove algebre, prsteni, grupe itd.), već su to samo algebre istog tipa. Klasična jednakost zamenjena je posebnom kompatibilnom rasplinutom (mrežno-vrednosnom) relacijom ekvivalencije. Na navedeni nacin i u cilju koji je u osnovi teze (poredenja sa postojecim pristupima u ovoj naucnoj oblasti) proucavane su (vec denisane)&nbsp; Ω-grupe. U nasim istraživanju uvedene su odgovarajuće normalne podgrupe. Uspostavljena je i istražena njihova veza sa Ω-kongruencijama. Normalna podgrupa&nbsp; Ω-grupe definisana je kao posebna&nbsp; klasa Ω-kongruencije. Jedan od rezultata u ovom delu je da su količničke grupe definisane pomocu nivoa Ω-jednakosti klasične normalne podgrupe odgovarajućih količničkih podgrupa polazne&nbsp; -grupe. I u ovom slučaju osnovna&nbsp; struktura na kojoj je denisana Ω-grupa je grupoid, ne nužno grupa. Opisane su osobine najmanje normalne podgrupe u terminima Ω-kongruencija, a date su i neke konstrukcije&nbsp; Ω-kongruencija.</p><p>Rezultati koji su izloženi u nastavku povezuju različite pristupe nekim mrežno- vrednosnim strukturama. Ω-Bulova algebra je uvedena na strukturi sa dve binarne, unarnom i dve nularne operacije, ali za koju se ne zahteva ispunjenost klasičnih aksioma. Identiteti za Bulove algebre važe kao mrežno-teoretske formule u odnosu na mrežno-vrednosnu jednakost. Klasicne Bulove algebre ih zadovoljavaju, ali obratno ne vazi: iz tih formula ne slede standardne aksiome za Bulove algebre. Na analogan nacin uveden je i&nbsp; Ω-Bulov prsten. Glavna svojstva ovih struktura su opisana. Osnovna osobina je da se klasične Bulove algebre odnosno Bulovi prsteni javljaju kao količničke strukture na nivoima Ω -jednakosti. Veza ove strukture sa Ω-Bulovom mrežom je pokazana.</p><p>Kao ilustracija ovih istraživanja, u radu je navedeno vi&scaron;e primera.</p>

Cette thèse est divisée en deux parties : la partie I traite de combinatoire additive, la partie II s’est portée sur des questions de théorie élémentaire des nombres. Dans le chapitre 1, on généralise la transformée de Davenport pour prouver que si S\mathbb A=(A, +)S est un demi-groupe cancellatif (éventuellement non commutatif) et SX, YS sont des sous-ensembles non vides de SAS tels que le sous semi groupe engendré par SYS est commutatif, on a SS|X+Y|\gc\min(\gamma(Y, |X|+|Y|-I)SS, où S\gamma(\ctlot)S dénote la constante de Cauchy-Davenport d’un ensemble. On en obtient une extension des théorèmes de Chowla et Pillai pour les groupes cycliques et une version plus forte d’un théorème additif de Karolyi et Hamidoune. Dans le chapitre 2, on montre que si S(A,+)S est un semi-groupe cancellatif et si SX, Y\subsetcq AS alors SS|X+Y|\gc\min(\gammaX+Y), |X|+|Y|-I)SS. Cela donne une généralisation de l’inégalité de Kemperman pour les groupes sans torsion et une version plus forte du théorème d’Hamidoune-Karolyi. Dans le chapitre 3, on généralise des résultats par Freiman et al., en prouvant que si S(A,\ctlot)S est un semi-groupe linéairement ordonnable et SSS est un sous-ensemble fini de SAS engendrant un sous-semi-groupe non-abélien, alors S|S^2-\gc3|S|-2S. Dans le chapitre 4, on prouve des résultats liés à une conjecture par Gyorgy et Smyth sur la finitude des entiers Sn\gc1S tels que Sn^kS divise Sa^a \pmb^nS pour des entiers fixés SaS, SbS et SkS avec Sk\gc3S, S|ab|\gc2Set S\gcd(a,b) = 1S. Enfin, dans le chapitre 5, on considère une question de divisibilité dans les entiers, en quelque sorte liée au problème de Znam et à la conjecture d’Agoh-Giuga / This thesis is divided into two parts. Part I is about additive combinatorics. Part II deals with questions in elementary number theory. In Chapter 1, we generalize the Davenport transform to prove that if si S\mathbb A=(A, +)S is acancellative semigroup (either abelian or not) and SX, YS are non-empty subsets of SAS such that the subsemigroup generated by SYS is abelian, then SS|X+Y|\gc\min(\gamma(Y, |X|+|Y|-I)SS, where for SZ\subsetcq AS we let S\gamma(Z):=\sup_{z_0\in Z^\times}\in f_(z_0\nc z\inZ) (vm ord)(z-z_0)S. This implies an extension of Chowla’s and Pillai’s theorems for cyclic groups and a stronger version of an addition theorem by Hamidoune and Karolyi for arbitrary groups. In Chapter 2, we show that if S(A, +) is a cancellative semigroup and SX, Y\subsetcq AS then SS|X+Y|\gc\min(\gammaX+Y), |X|+|Y|-I)SS. This gives a generalization of Kemperman’s inequality for torsion free groups and a stronger version of the Hamidoune-Karolyi theorem. In Chapter 3, we generalize results by Freiman et al. by proving that if S(A,\ctlot)S is a linearly orderable semigroup and SSS is a finite subset of SAS generating a non-abelian subsemigroup, then S|S^2-\gc3|S|-2S. In Chapter 4, we prove results related to conjecture by Gyory and Smyth on the sets SR_k^\pm(a,b)S of all positive integers SnS such that Sn^kS divides Sa^a \pmb^nS for fixed integers SaS, SbS and SkS with Sk\gc3S, S|ab|\gc2Set S\gcd(a,b) = 1S. In particular, we show that SR_k^pm(a,b)S is finite if Sk\gc\max(|a|.|b|)S. In Chapter 5, we consider a question on primes and divisibility somchow related to Znam’s problem and the Agoh-Giuga conjecture

This thesis is divided into two parts. Part I is about additive combinatorics. Part II deals with questions in elementary number theory. In Chapter 1, we generalize the Davenport transform to prove that if si S\mathbb A=(A, +)S is acancellative semigroup (either abelian or not) and SX, YS are non-empty subsets of SAS such that the subsemigroup generated by SYS is abelian, then SS|X+Y|\gc\min(\gamma(Y, |X|+|Y|-I)SS, where for SZ\subsetcq AS we let S\gamma(Z):=\sup_{z_0\in Z^\times}\in f_(z_0\nc z\inZ) (vm ord)(z-z_0)S. This implies an extension of Chowla's and Pillai's theorems for cyclic groups and a stronger version of an addition theorem by Hamidoune and Karolyi for arbitrary groups. In Chapter 2, we show that if S(A, +) is a cancellative semigroup and SX, Y\subsetcq AS then SS|X+Y|\gc\min(\gammaX+Y), |X|+|Y|-I)SS. This gives a generalization of Kemperman's inequality for torsion free groups and a stronger version of the Hamidoune-Karolyi theorem. In Chapter 3, we generalize results by Freiman et al. by proving that if S(A,\ctlot)S is a linearly orderable semigroup and SSS is a finite subset of SAS generating a non-abelian subsemigroup, then S|S^2-\gc3|S|-2S. In Chapter 4, we prove results related to conjecture by Gyory and Smyth on the sets SR_k^\pm(a,b)S of all positive integers SnS such that Sn^kS divides Sa^a \pmb^nS for fixed integers SaS, SbS and SkS with Sk\gc3S, S|ab|\gc2Set S\gcd(a,b) = 1S. In particular, we show that SR_k^pm(a,b)S is finite if Sk\gc\max(|a|.|b|)S. In Chapter 5, we consider a question on primes and divisibility somchow related to Znam's problem and the Agoh-Giuga conjecture

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The advent in 1801 of the brilliant work Disquisitiones Arithmeticae of Carl Gauss Friedrich (1777-1885) provided extremely important elements to Number Theory, including the study of congruences, which attracts the eyes of many mathematicians to this day for its application in various areas, including basic education issues, highlighting the need of their study as algebraic learning tool. Therefore, this paper proposes to approach the study of congruence in a systematic way, in order to the context of basic education, by proposing didactic sequence, and their application to everyday problems. / O advento, em 1801, da brilhante obra Disquisitiones Arithmeticae} de Carl Friedrich Gauss (1777-1885) proporcionou elementos de extraordinária importância para a Teoria dos Números, entre eles o estudo de congruência, o qual atrai os olhares de diversos matemáticos até os dias atuais pela sua aplicação em diversas áreas, inclusive em temas do ensino básico, evidenciando a necessidade do seu estudo como ferramenta de aprendizagem algébrica. Sendo assim, o presente trabalho propõe abordar o estudo de congruência de forma sistemática, visando a sua contextualização na educação básica, através de uma proposta de sequência didática, e sua aplicação em problemas do cotidiano.

The purpose of this dissertation is to present to the students of basic education a powerful tool in the resolution of Arithmetic such as Modular Congruence. We initiate our study by approaching the main basics concepts of Number Theory: Divisibility, Eucledian Division, Greatest Common Divisor, Remainder modular arytmetics, culminating with Modular Congruence and its applications: Chinese Remainder Theorem and Intergers. / A presente dissertação tem como objetivo apresentar aos alunos do ensino básico uma poderosa ferramenta na resolução de problemas aritméticos, que é a Congruência modular. Para tanto, iniciamos nosso estudo abordando conceitos básicos da teoria dos números: divisibilidade, divisão euclidiana, máximo divisor comum, mínimo múltiplo comum, análise de restos, culminando com a congruência modular e algumas de suas aplicações: Teorema Chinês dos restos e Partilha de senhas.

No description available.

Euclidean Geometry is a challenging topic for most of the learners in the secondary schools. A qualitative case study explores the reasoning skills of ninth graders in the proving of congruent triangles in their natural environment. A class of thirty-two learners was conveniently selected to participate in the classroom observations. Two groups of six learners each were purposefully selected from the same class of thirty-two learners to participate in focus group interviews. The teaching documents were analysed. The Van Hiele’s levels of geometric thinking were used to reflect on the reasoning skills of the learners. The findings show that the majority of the learners operated at level 2 of Van Hiele’s geometric thinking. The use of visual aids in the teaching of geometry is important. About 30% of the learners were still operating at level 1 of Van Hiele theory. The analysed books showed that investigation help learners to discover the intended knowledge on their own. Learners need quality experience in order to move from a lower to a higher level of Van Hiele’s geometry thinking levels. The study brings about unique findings which may not be generalised. The results can only provide an insight into the reasoning skills of ninth graders in proving of congruent triangles. I recommend that future researchers should focus on proving of congruent triangles with a bigger sample of learners from different environmental settings. / Mathematics Education / M. Ed. (Mathematics Education)