Dokazi prebrojavanjem

master's thesis

• Author: Helena Štimac
• Mentor: Vedran Krčadinac (mentor)
• Committee member: Vedran Krčadinac (predsjednik povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Committee member: Nina Kamčev (član povjerenstva)
• Committee member: Goran Igaly (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-11-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U radu se kroz tri poglavlja upoznajemo s kombinatornim interpretacijama Fibonaccijevih brojeva, verižnih razlomaka te binomnih koeficijenata i kombinacija s ponavljanjem. Poglavlje o Fibonaccijevim brojevima posebno približava tehnike dokazivanja prebrojavanjem te služi kao temelj u razmišljanju o daljnjim poglavljima i složenijim problemima. Iskazano je mnoštvo identiteta. Svaki identitet dokazujemo na elegantan način temeljen na konceptu prebrojavanja. U radu dokaze provodimo na dva načina,... More dvostrukim prebrojavanjem istog skupa ili prebrojavanjem dva različita skupa, a zatim uspostavljanjem bijekcije između njih. U oba slučaja slijedit će jednakost između lijeve i desne strane identiteta. Iako nije uvijek najjednostavniji pristup, prebrojavanje nam omogućuje potpuno drugačije shvaćanje i razumijevanje problema. Ovim radom želimo čitatelju dočarati moć prebrojavanja kada je u pitanju dokazivanje kombinatornih identiteta. Less

Abstract (english)

In this thesis, through three chapters, we introduce combinatorial interpretations of Fibonacci numbers, continued fractions, binomial coefficients and combinations with repetition. In the chapter on Fibonacci numbers we get acquainted with proof techniques by counting. It serves as a foundation for further chapters and more complex problems. Many identities are expressed. We give an elegant proof based on counting for each identity. In this thesis, counting arguments are carried out in two... More ways, by double counting the same set or by counting two different sets and then establishing a bijection between them. In both cases, equality between the left and right side of the identity will follow. Although it is not always the simplest approach, counting allows us to see and understand the problem in a completely different way. With this thesis, we want to convey to the reader the power of counting arguments when it comes to proving combinatorial identities. Less

• Keywords: Fibonaccijevi brojevi; verižni razlomci; binomni koeficijenti; kombinacije s ponavljanjem; kombinatorne interpretacije; identiteti; tehnike dokazivanja prebrojavanjem; dvostruko prebrojavanje skupa; uspostavljanje bijekcije između dvaju skupova
• Keywords (english): Fibonacci numbers; continued fractions; binomial coefficients; combinations with repetition; combinatorial identities; proof techniques by counting; double counting the same set; establishing a bijection between two sets
• Language: croatian
• URN:NBN: urn:nbn:hr:217:100418
• Study programme: Title: Financial and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-11-11 13:22:02