On the Lattice of Intervals and Rough Sets

Adam Grabowski; Magdalena Jastrzębska

Formalized Mathematics (2009)

  • Volume: 17, Issue: 4, page 237-244
  • ISSN: 1426-2630

Abstract

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Rough sets, developed by Pawlak [6], 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 [3], is also a step towards the formalization of the algebraic theory of rough sets, as in [4] or [9].

How to cite

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Adam Grabowski, and Magdalena Jastrzębska. "On the Lattice of Intervals and Rough Sets." Formalized Mathematics 17.4 (2009): 237-244. <http://eudml.org/doc/267013>.

@article{AdamGrabowski2009,
abstract = {Rough sets, developed by Pawlak [6], 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 [3], is also a step towards the formalization of the algebraic theory of rough sets, as in [4] or [9].},
author = {Adam Grabowski, Magdalena Jastrzębska},
journal = {Formalized Mathematics},
language = {eng},
number = {4},
pages = {237-244},
title = {On the Lattice of Intervals and Rough Sets},
url = {http://eudml.org/doc/267013},
volume = {17},
year = {2009},
}

TY - JOUR
AU - Adam Grabowski
AU - Magdalena Jastrzębska
TI - On the Lattice of Intervals and Rough Sets
JO - Formalized Mathematics
PY - 2009
VL - 17
IS - 4
SP - 237
EP - 244
AB - Rough sets, developed by Pawlak [6], 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 [3], is also a step towards the formalization of the algebraic theory of rough sets, as in [4] or [9].
LA - eng
UR - http://eudml.org/doc/267013
ER -

References

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  1. [1] Grzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719-725, 1991. 
  2. [2] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990. 
  3. [3] Adam Grabowski. Basic properties of rough sets and rough membership function. Formalized Mathematics, 12(1):21-28, 2004. 
  4. [4] Amin Mousavi and Parviz Jabedar-Maralani. Relative sets and rough sets. Int. J. Appl. Math. Comput. Sci., 11(3):637-653, 2001. Zbl0986.03042
  5. [5] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990. 
  6. [6] Z. Pawlak. Rough sets. International Journal of Parallel Programming, 11:341-356, 1982, doi:10.1007/BF01001956.[Crossref] Zbl0501.68053
  7. [7] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990. 
  8. [8] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  9. [9] Y. Y. Yao. Interval-set algebra for qualitative knowledge representation. Proc. 5-th Int. Conf. Computing and Information, pages 370-375, 1993. 
  10. [10] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215-222, 1990. 

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