DSpace Kolekcja: http://hdl.handle.net/11320/5549 2020-11-24T18:13:37Z Niven’s Theorem http://hdl.handle.net/11320/5564 Tytu&#322;: Niven’s Theorem Autorzy: Korniłowicz, Artur; Naumowicz, Adam Abstrakt: This article formalizes the proof of Niven’s theorem  which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9]. 2016-01-01T00:00:00Z The Axiomatization of Propositional Logic http://hdl.handle.net/11320/5562 Tytu&#322;: The Axiomatization of Propositional Logic Autorzy: Giero, Mariusz Abstrakt: This article introduces propositional logic as a formal system (, , ). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes α ⇒ (β ⇒ α),(α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)),(¬β ⇒ ¬α) ⇒ ((¬β ⇒ α) ⇒ β). Modus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved. The proof of the completeness theorem is carried out by a counter-model existence method. In order to prove the completeness theorem, Lindenbaum’s Lemma is proved. Some most widely used tautologies are presented. 2016-01-01T00:00:00Z Algebraic Numbers http://hdl.handle.net/11320/5563 Tytu&#322;: Algebraic Numbers Autorzy: Watase, Yasushige Abstrakt: This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers , ,  and  are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to the polynomial ring of ℚ[x] turns to be a field. 2016-01-01T00:00:00Z On Subnomials http://hdl.handle.net/11320/5560 Tytu&#322;: On Subnomials Autorzy: Ziobro, Rafał Abstrakt: While discussing the sum of consecutive powers as a result of division of two binomials W.W. Sawyer  observes “It is a curious fact that most algebra textbooks give our ast result twice. It appears in two different chapters and usually there is no mention in either of these that it also occurs in the other. The first chapter, of course, is that on factors. The second is that on geometrical progressions. Geometrical progressions are involved in nearly all financial questions involving compound interest – mortgages, annuities, etc.” It’s worth noticing that the first issue involves a simple arithmetical division of 99...9 by 9. While the above notion seems not have changed over the last 50 years, it reflects only a special case of a broader class of problems involving two variables. It seems strange, that while binomial formula is discussed and studied widely , , little research is done on its counterpart with all coefficients equal to one, which we will call here the subnomial. The study focuses on its basic properties and applies it to some simple problems usually proven by induction . 2016-01-01T00:00:00Z