Egzistencija realnih brojeva

master's thesis

• Author: Vedran Levanić
• Mentor: Zvonko Iljazović (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Committee member: Vedran Čačić (član povjerenstva)
• Committee member: Tomislav Berić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj diplomski rad podijeljen je na pet većih tematskih poglavlja kroz koje se provodi konstrukcija skupa realnih brojeva, odnosno jednog potpuno uređneog polja. Prvo poglavlje sadrži definicije osnovnih pojmova. To su pojmovi binarnih operacija i binarnih relacija na nepraznom skupu i njihova svojstva te pojmovi različitih algebarskih struktura. U drugom poglavlju opisani su aksiomi prirodnih brojeva, definirane operacije zbrajanja i množenja na skupu prirodnih brojeva uz njihova svojstva te konstruiran uređaj na istom skupu. Skup cijelih brojeva, operacije zbrajanja i množenja te uređaj na njemu definirani su u trećem poglavlju. Četvrto poglavlje sadrži konstrukciju skupa racionalnih brojeva te već spomenute pojmove uz dodatnu izgradnju koncepta apsolutne vrijednosti apsolutnog broja. Peto poglavlje započinje pojmom niza i njegove konvergencije. Uz definiciju Cauchyjeva niza u skupu racionalnih brojeva i relacije ekvivalencije na skupu Cauchyjevih nizova dolazi se do definicije skupa realnih brojeva. Nakon toga, uz izgradnju operacija zbrajanja i množenja te uređaja na skupu realnih brojeva, preko Cauchyjevih nizova na tom skupu i dokaza da su oni konvergentni dolazi se do konačnog koncepta potpuno uređenog polja.

Abstract (english)

This graduate thesis is divided into five major thematic chapters through which the construction of a set of real numbers, that is, a completely ordered field, is shown. The first chapter contains the definitions of basic notions. These are the notions of binary operations and binary relations on a nonempty set and their properties, as well as the concepts of various algebraic structures. In the second chapter we give the axioms of natural numbers, we define addition and multiplication on the set of natural numbers, we prove their properties and construct an order on the same set. The set of integers, addition, multiplication and the order on it are defined in the third chapter. The fourth chapter contains the construction of the set of rational numbers and the aforementioned concepts with the additional concept of the absolute value of a rational number. The fifth chapter begins with the concept of a sequence and its convergence. With the definition of Cauchy sequence in the set of rational numbers and the relation of equivalence at the Cauchy sequence, a definition of a set of real numbers comes up. Thereafter, along with the construction of the addition, multiplication and the order on the set of real numbers, through Cauchy sequences in that set and the proof that they are convergent, comes the ultimate concept of a completely ordered field.

• Keywords: konstrukcija skupa realnih brojeva; algebarske strukture; konstrukcija skupa racionalnih brojeva; operacije na skupu; apsolutna vrijednost apsolutnog broja; Cauchyjev niz; koncept potpuno uređenog polja
• Keywords (english): construction of a set of real numbers; algebraic structures; construction of the set of rational numbers; addition and multiplication on sets; absolute value of a rational number; Cauchy sequence; concept of a completely ordered field
• Language: croatian
• URN:NBN: urn:nbn:hr:217:817866
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2018-02-02 12:12:33