DSpace Zesp&#243;&#322;: http://hdl.handle.net/11320/20 2020-09-25T13:08:17Z Introduction to Formal Preference Spaces http://hdl.handle.net/11320/3691 Tytu&#322;: Introduction to Formal Preference Spaces Autorzy: Niewiadomska, Eliza; Grabowski, Adam Abstrakt: In the article the formal characterization of preference spaces  is given. As the preference relation is one of the very basic notions of mathematical economics , it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done - see ). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished . There are many approaches to preferences in literature. We modelled them in a rather illustrative way (similar structures were considered in ): either the consumer (strictly) prefers an alternative, or they are of equal interest; he/she could also have no opinion of the choice. Then our structures are based on three relations on the (arbitrary, not necessarily finite) set of alternatives. The completeness property can however also be modelled, although we rather follow  which is more general . Additionally we assume all three relations are disjoint and their set-theoretic union gives a whole universe of alternatives. We constructed some positive and negative examples of preference structures; the main aim of the article however is to give the characterization of consumer preference structures in terms of a binary relation, called characteristic relation , and to show the way the corresponding structure can be obtained only using this relation. Finally, we show the connection between tournament and total spaces and usual properties of the ordering relations. 2013-01-01T00:00:00Z Representation of the Fibonacci and Lucas Numbers in Terms of Floor and Ceiling http://hdl.handle.net/11320/3562 Tytu&#322;: Representation of the Fibonacci and Lucas Numbers in Terms of Floor and Ceiling Autorzy: Jastrzębska, Magdalena Abstrakt: In the paper we show how to express the Fibonacci numbers and Lucas numbers using the floor and ceiling operations. 2010-01-01T00:00:00Z On the Lattice of Intervals and Rough Sets http://hdl.handle.net/11320/3550 Tytu&#322;: On the Lattice of Intervals and Rough Sets Autorzy: Grabowski, Adam; Jastrzębska, Magdalena Abstrakt: Rough sets, developed by Pawlak , are an important tool to describe a situation of incomplete or partially unknown information. One of the algebraic models deals with the pair of the upper and the lower approximation. Although usually the tolerance or the equivalence relation is taken into account when considering a rough set, here we rather concentrate on the model with the pair of two definable sets, hence we are close to the notion of an interval set. In this article, the lattices of rough sets and intervals are formalized. This paper, being essentially the continuation of , is also a step towards the formalization of the algebraic theory of rough sets, as in  or . 2009-01-01T00:00:00Z