Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

**Adam Grabowski**- University of Bialystok

- We present basic concepts concerning rough set theory. We define tolerance and approximation spaces and rough membership function. Different rough inclusions as well as the predicate of rough equality of sets are also introduced.

This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

- Preliminaries
- Tolerance and Approximation Spaces
- Membership Function
- Rough Inclusion
- Rough Equality of Sets

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Jozef Bialas.
Group and field definitions.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Basic functions and operations on functions.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [10]
Czeslaw Bylinski.
The sum and product of finite sequences of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [11]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Adam Grabowski.
On the category of posets.
*Journal of Formalized Mathematics*, 8, 1996. - [13]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Andrzej Nedzusiak.
$\sigma$-fields and probability.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Andrzej Nedzusiak.
Probability.
*Journal of Formalized Mathematics*, 2, 1990. - [16]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Konrad Raczkowski and Pawel Sadowski.
Equivalence relations and classes of abstraction.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [19]
Andrzej Trybulec.
Binary operations applied to functions.
*Journal of Formalized Mathematics*, 1, 1989. - [20]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [21]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [22]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [23]
Wojciech A. Trybulec.
Partially ordered sets.
*Journal of Formalized Mathematics*, 1, 1989. - [24]
Wojciech A. Trybulec.
Groups.
*Journal of Formalized Mathematics*, 2, 1990. - [25]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [26]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [27]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989. - [28]
Edmund Woronowicz and Anna Zalewska.
Properties of binary relations.
*Journal of Formalized Mathematics*, 1, 1989.

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