Izračunljivost na skupovima $\mathbb{Z}, \mathbb{Q}, \mathbb{R}$ i $\mathbb{C}$

Posavčević, Ivan (2016) Izračunljivost na skupovima $\mathbb{Z}, \mathbb{Q}, \mathbb{R}$ i $\mathbb{C}$. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

This thesis is logically divided into two chapters. The first chapter ”Countable computability” starts with basic concepts and terms associated with computability on $\mathbb{N}$. Using these concepts and terms, there are introduced analogous terms for countable supersets of $\mathbb{N}$ and detail proven the computability of certain functions. It is spoken about the term of set dimension and the representation of supersets of $\mathbb{N}$ using the natural numbers. The main result of the chapter is the proof of computable analogue of the reminder theorem. The second chapter ”Uncountable computability” covers the foundations of computability on $\mathbb{R}$ and one example of computability on $\mathbb{C}$ in detail. The significant difference in comparison to the previous chapter lies in representation of the real number, which brings a big turnaround in the approach to the problem of computability. However, terms from discrete case are used profusely and in that way these two differently motivated approaches are merged into a coherent unit. The main result of the whole thesis is the proof of computability of quadratic equation’s solutions. Since the thesis is written in the form of scripts, it could be used in the case of appearance of (elective) courses associated with this topic.

Item Type: Thesis (Diploma thesis)
Supervisor: Čačić, Vedran
Date: 2016
Number of Pages: 37
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 09 Nov 2016 13:43
Last Modified: 09 Nov 2016 13:43
URI: http://digre.pmf.unizg.hr/id/eprint/5277

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