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A Geometry of ApproximationLocal Validity, Grothendieck Topologies and Rough Sets

A Geometry of Approximation: Local Validity, Grothendieck Topologies and Rough Sets Chapter 7 Local Validity, Grothendieck Topologies and Rough Sets 7.1 Representing Rough Sets The first step is to represent rough sets. Thus, we now give the formal definition of a rough set and the formal definition of the decreasing representation of rough sets which was adopted in the Introduction. Definition 7.1.1. Given an Indiscernibility Space U, E, 1. Two sets X, Y ∈ ℘(U ) are called rough top equal, X 0 Y ,iff (uE)(X)= (uE)(Y ). 2. Two sets X, Y ∈ ℘(U ) are called rough bottom equal, X∼Y ,iff (lE)(X)= (lE)(Y ). 3. Two sets X, Y ∈ ℘(U ) are called rough equal, X ≈ Y ,iff X 0 Y and X ∼ Y . 4. A set X ∈ ℘(U ) is called definable iff X =(lE)(X)= (uE)(X). 5. A set X ∈ ℘(U ) is called undefinable iff (lE)(X)= ∅ and (uE)(X)= U . 6. Any equivalence class of subsets of U modulo the relation ≈ is called a rough set. 211 212 7 Local Validity, Grothendieck Topologies and Rough Sets We represent rough sets, with respect to an Indiscernibility Space U, E by means of ordered pairs of the form (uE)(X), (lE)(X),and http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Geometry of ApproximationLocal Validity, Grothendieck Topologies and Rough Sets

Part of the Trends in Logic Book Series (volume 27)
Editors: Pagliani, Piero; Chakraborty, Mihir
A Geometry of Approximation — Jan 1, 2008
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Publisher
Springer Netherlands
Copyright
© Springer Netherlands 2008
ISBN
978-1-4020-8621-2
Pages
211 –236
DOI
10.1007/978-1-4020-8622-9_7
Publisher site
See Chapter on Publisher Site

Abstract

Chapter 7 Local Validity, Grothendieck Topologies and Rough Sets 7.1 Representing Rough Sets The first step is to represent rough sets. Thus, we now give the formal definition of a rough set and the formal definition of the decreasing representation of rough sets which was adopted in the Introduction. Definition 7.1.1. Given an Indiscernibility Space U, E, 1. Two sets X, Y ∈ ℘(U ) are called rough top equal, X 0 Y ,iff (uE)(X)= (uE)(Y ). 2. Two sets X, Y ∈ ℘(U ) are called rough bottom equal, X∼Y ,iff (lE)(X)= (lE)(Y ). 3. Two sets X, Y ∈ ℘(U ) are called rough equal, X ≈ Y ,iff X 0 Y and X ∼ Y . 4. A set X ∈ ℘(U ) is called definable iff X =(lE)(X)= (uE)(X). 5. A set X ∈ ℘(U ) is called undefinable iff (lE)(X)= ∅ and (uE)(X)= U . 6. Any equivalence class of subsets of U modulo the relation ≈ is called a rough set. 211 212 7 Local Validity, Grothendieck Topologies and Rough Sets We represent rough sets, with respect to an Indiscernibility Space U, E by means of ordered pairs of the form (uE)(X), (lE)(X),and

Published: Jan 1, 2008

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