diplomsko delo
Tina Orejaš (Author), Marko Slapar (Mentor)

Abstract

Množica je števno neskončna, če je ekvipolentna (ima enako moč) množici naravnih števil. Števna neskončnost je najmanjša neskončnost, v smislu, da ima vsaka neskončna množica števno neskončno podmnožico. Množica realnih števil ni števno neskončna, kar klasično dokažemo s protislovjem, če predpostavimo, da obstaja surjekcija iz množice naravnih števili v množico realnih števil. Obstaja pa tudi alternativni dokaz s pomočjo neskončne igre realnih števil. Pri igri imamo dva igralca, ki si najprej izbereta neko podmnožico S intervala [0,1], nato pa izmenjujoče izbirata realna števila. Prvi igralec izbere neko število a_1 med 0 in 1. Drugi igralec potem izbere neko število b_1 med a_1 in 1. Tako v n-tem krogu prvi igralec izbere realno število a_n, za katero velja a_(n-1)≤a_n≤b_(n-1), potem pa drugi igralec izbere število b_n, tako da velja a_n≤b_n≤b_(n-1). Prvi igralec ima zmagovalno strategijo, če lahko, ne glede na strategijo drugega igralca, števila vedno izbira tako, da je α=lim┬(n→∞)⁡〖a_n 〗 v množici S (vsako naraščajoče zaporedje realnih števil, ki je navzgor omejeno, ima limito). Če je množica S kar interval [0,1], ima prvi igralec seveda zmagovalno strategijo, malo težje pa je videti, da prvi igralec nima zmagovalne strategije, če je množica S števna.

Keywords

realna števila;Cantor;Cantorjeva igra;množice;neštevnost;Borelove množice;

Data

Language: Slovenian
Year of publishing:
Typology: 2.11 - Undergraduate Thesis
Organization: UL PEF - Faculty of Education
Publisher: [T. Orejaš]
UDC: 511.11(043.2)
COBISS: 11705929 Link will open in a new window
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Downloads: 161
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Other data

Secondary language: English
Secondary title: Uncountable sets and an real number game
Secondary abstract: A set is countably infinite, if it is equipolent (has the same cardinality), as the set of natural numbers. Countable infinity is the smallest infinity, meaning that every infinite set has a countably infinite subset. The set of real numbers is not countably infinite, which is usually proved by contradiction, if we assume, that there exist a surjection from the set of natural numbers to the set of real numbers. Beside the classic proof, there exist an alternative proof with the help of an infinite real number game. Two players choose some subset S of interval [0,1], and then they alternate choosing real numbers. The first player chooses any real number a_1 between 0 and 1. The second player then chooses any real number b_1 between a_1 and 1. In round n the first player chooses any real number a_n, which satisfies the condition a_(n-1)≤a_n≤b_(n-1), and then the second player chooses number b_n, so that a_n≤b_n≤b_(n-1). The first player has a winning strategy, if he can, without considering the other player strategy, choose the numbers so that α=lim┬(n→∞)⁡〖a_n 〗 is in the set S (because every ascending sequence of real numbers, which is limited above, has a limit). If set S is equivalent to the interval [0,1], the first player has winning strategy, but it is harder to see, that the first player doesn't have a winning strategy, if the set S is countable.
Secondary keywords: mathematics;matematika;
File type: application/pdf
Type (COBISS): Bachelor thesis/paper
Thesis comment: Univ. v Ljubljani, Pedagoška fak., Dvopredmetni učitelj
Pages: 20 str.
ID: 10864599
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