Catoptrical rough set model on two universes using granule-based definition and its variable precision extensions

Generalized Variable Precision Rough Sets With the Inclusion Errors

“…Definition 2.1. [3,17] Let P (X) = {A | A ⊂ X} be given. A map e : P (X) × P (X) → [0, 1] is called an inclusion error defined by…”

“…The pair (A, A) is called a generalized rough set of A. As an extension of Ziarko's rough sets, Dai et al [3] introduced the generalized variable precision rough set (A, A) defined by…”

“…In this paper, we introduce the notion of generalized variable precision rough sets with the inclusion errors as an extension of [3,17]. We investigate the properties of generalized upper approximation operator R β and generalized lower approximation operator R…”

New Operations of Picture Fuzzy Relations and Fuzzy Comprehensive Evaluation

“…Most of them are vague than precise, and uncertainty is a common research topic of many branches of science (economics, engineering, environment, management science, medical science, and so on). However, uncertainty is an unintelligible expression without a straightforward description, and many theories were established, such as probability theory, fuzzy set theory [1][2][3], intuitionistic fuzzy set theory [4], hesitant fuzzy set theory [5][6][7][8], soft set theory [9][10][11][12], rough set theory [13][14][15][16][17], granular computing [18][19][20][21][22][23][24][25][26][27], et al…”