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Chapter 8 Approximation and Algebraic Logic 8.1 Approximation Operators Of course, in order to honour the philosophy of Rough Set Theory, we must endow any Rough Set Systems RS(U ) with a couple of operators reﬂecting the approximation features provided by the theory, applied, this time, not to sets but to rough sets. Let us connect the knowledge- oriented interpretation of these operators to their logical properties. We are looking for two operators M and L such that the following diagrams commute: (uE) (lE) ℘(U ) AS(U ) ℘(U ) AS(U ) rs rs rs rs RS(U ) rs(AS(U )) RS(U ) rs(AS(U )) M L Proposition 8.1.1. If the above diagrams commute then for any rough set (uE)(X), (lE)(X), 1. L((uE)(X), (lE)(X))= (lE)(X), (lE)(X). 2. M ((uE)(X), (lE)(X))= (uE)(X), (uE)(X). Proof. For any X ∈ ℘(U ), (lE)(X) ∈ AS(U ). But for any element A of AS(U ), rs(A)= A, A.Thus rs((lE)(X)) = (lE)(X), (lE)(X). Similarly for M . qed 237 238 8 Approximation and Algebraic Logic So, let us deﬁne the two operators in the following manner: Deﬁnition 8.1.1. For any Rough Set System RS(U ), for any a ∈ RS(U ), (1) M (a)= a
Published: Jan 1, 2008
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