Abstract
In this paper, we investigate the following problem: give a quasi-Boolean function Ψ(x 1, …, x n ) = (a ∧ C) ∨ (a 1 ∧ C 1) ∨ … ∨ (a p ∧ C p ), the term (a ∧ C) can be deleted from Ψ(x 1, …, x n )? i.e., (a ∧ C) ∨ (a 1 ∧ C 1) ∨ … ∨ (a p ∧ C p ) = (a 1 ∧ C 1) ∨ … ∨ (a p ∧ C p )? When a = 1: we divide our discussion into two cases. (1) ℑ1(Ψ,C) = ø, C can not be deleted; ℑ1(Ψ,C) ≠ø, if S 0 i ≠ø (1 ≤ i ≤ q), then C can not be deleted, otherwise C can be deleted. When a = m: we prove the following results: (m∧C)∨(a 1∧C 1)∨…∨(a p ∧C p ) = (a 1∧C 1)∨…∨(a p ∧C p ) ⇔ (m ∧ C) ∨ C 1 ∨ … ∨C p = C 1 ∨ … ∨C p . Two possible cases are listed as follows, (1) ℑ2(Ψ,C) = ø, the term (m∧C) can not be deleted; (2) ℑ2(Ψ,C) ≠ø, if (∃i 0) such that \(S'_{i_0 } \) = ø, then (m∧C) can be deleted, otherwise ((m∧C)∨C 1∨…∨C q )(v 1, …, v n ) = (C 1 ∨ … ∨ C q )(v 1, …, v n )(∀(v 1, …, v n ) ∈ L n3 ) ⇔ (C ′1 ∨ … ∨ C ′ q )(u 1, …, u q ) = 1(∀(u 1, …, u q ) ∈ B n2 ).
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Cheng, Yj., Xu, Lx. Properties of quasi-Boolean function on quasi-Boolean algebra. Fuzzy Inf. Eng. 3, 275–291 (2011). https://doi.org/10.1007/s12543-011-0083-8
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DOI: https://doi.org/10.1007/s12543-011-0083-8