Topics in Rough Set Theory by Seiki Akama & Yasuo Kudo & Tetsuya Murai

Author:Seiki Akama & Yasuo Kudo & Tetsuya Murai
Language: eng
Format: epub
ISBN: 9783030295660
Publisher: Springer International Publishing

A Proofs of Theoretical Results

Lemma 5.1

The collection of sets defined as

comprises a partition of U.

Proof

From the definitions of support set , nonsupport set , and , , and , it is obvious that, for any , if , , and hold, then holds. Thus, it is sufficient to show . Moreover, every set is obviously a subset of U, which implies .

Let be any object. If both and hold, either or holds, which implies that either or holds. Otherwise, if or or both hold, either or or hold. These facts imply that ; thus, .

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