Elementarni aspekti potpunosti

master's thesis

• Title (english): Elementary aspects of completeness
• Author: Iva Ćorak
• Mentor: Zvonko Iljazović (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Igor Ciganović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-03-05, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj diplomski rad podijelili smo na tri veća poglavlja u kojima smo promatrali neke elementarne aspekte potpunosti. Prvo poglavlje sadrži definicije osnovnih definicija, kao što su binarna operacija, binarna relacija, polje i uređaj na skupu. Na kraju prvog pogljavlja proučili smo svojstva uređenog polja. U drugom poglavlju fiksirali smo jedno potpuno uređeno polje te u njemu konstruirali skupove \(\mathbb {N}\), \(\mathbb {Z}\) i \(\mathbb { Q }\). Proučili smo svojstva tih skupova a ujedno i omeđenost. U trećem poglavlju dokazali smo da ne postoji \(\sqrt{2}\) u \( \mathbb { Q }\) a zatim smo, koristeći taj rezultat, dokazali da \( \mathbb { Q }\) nije potpuno uređeno polje. Također smo pokazali da postoji \(\sqrt{2}\) u \( \mathbb { R }\) .Na kraju smo promatrali nizove i konvergenciju nizova u uređenim poljima.

Abstract (english)

This graduate thesis is divided into three chapters through which we shown some elementary aspects of completeness. First chapter contains the basic definitions, such as binary operation, binary relation, field and order on a set. At the end of first chapter we studied some properties of an ordered field. In second chapter we fixed one complete ordered field and we constructed sets \(\mathbb {N}\), \(\mathbb {Z}\) i \(\mathbb { Q }\). We studied properties of those sets and at the same time we studied bounded sets. In third chapter we proved that there is no \(\sqrt{2}\) in \( \mathbb { Q }\) and then, using that result, we proved that \( \mathbb { Q }\) is not a complete ordered field. We also showed that there is \(\sqrt{2}\) in \( \mathbb { R }\). At the end we studied a sequence and convergence of a sequence in an ordered field.

• Keywords: binarna operacija; binarna relacija; polje; uređaj na skupu
• Keywords (english): binary operation; binary relation; field; order on a set
• Language: croatian
• URN:NBN: urn:nbn:hr:217:394596
• 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: 2021-03-24 09:58:24