Konačnost i aksiomi prirodnih brojeva

master's thesis

• Author: Kristina Milković
• Mentor: Zvonko Iljazović (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Committee member: Dragutin Svrtan (član povjerenstva)
• Committee member: Zvonimir Tutek (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-07-18, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu proučavali smo skup prirodnih brojeva te pojam konačnog i prebrojivog skupa. U prvom smo poglavlju, krenuvši od aksioma, uveli osnovne pojmove vezane uz skup prirodnih brojeva kao što su zbrajanje prirodnih brojeva i uređaj na skupu prirodnih brojeva te smo dokazali neke činjenice vezane uz to. Također, iskazali smo i dokazali princip definicije indukcijom. U drugom smo poglavlju uveli pojam konačnog skupa te dokazali osnovne činjenice vezane uz to. Nadalje, proučavali smo beskonačne skupove te dokazali da je skup beskonačan ako i samo ako je ekvipotentan svom pravom podskupu. Pojam prebrojivog skupa je bio središnji pojam trećeg poglavlja. Ispitivali smo razna svojstva ovoga pojma te smo dokazali da je unija prebrojive familije prebrojivih skupova i sama prebrojiva. Navedeni pojmovi imaju važnu ulogu u teoriji skupova, što je prikazano i u ovom diplomskom radu.

Abstract (english)

In this thesis we studied a set of natural numbers and the concept of a finite and a countable set. In the first chapter, starting with the axioms, we introduced basic concepts related to a set of natural numbers such as the addition of natural numbers and the order on the set of natural numbers, and we have proven the facts related to it. We also demonstrated and proved the principle of definition by induction. In the second chapter, we introduced the concept of a finite set and we proved the basic facts related to it. Furthermore, we have studied infinite sets and we proved that a set is infinite if and only if it is equipotent to its proper subset. The countable set was the central notion of the third chapter. We examined the various properties of this concept and we proved that the union of a countable family of countable sets is countable. These notions have significant role in the theory of sets, which is shown in this thesis.

• Keywords: konačni skup; prebrojivi skup; skup prirodnih brojeva; definicija indukcijom; beskonačan skup
• Keywords (english): finite set; countable set; set of natural numbers; definition by induction; infinite set
• Language: croatian
• URN:NBN: urn:nbn:hr:217:656283
• 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: 2018-03-01 11:42:00