By Li P.-F., Liu M.-Q., Zhang R.-C.

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**Extra info for 2^m 4^1 designs with minimum aberration or weak minimum aberration**

**Sample text**

26. Here, both Rod and Freddy belong to the (fuzzy) set of old people, but Rod has a higher degree of membership to this set. The speciﬁcation of membership functions is typically subjective, as can be seen in this example. There are many justiﬁable deﬁnitions of the concept Old . Indeed people of different ages may deﬁne this concept quite differently. One way of constructing the membership function is by using a voting model, and this way the generated fuzzy sets can be rooted in reality with clear semantics.

Am to a given entry of the discernibility matrix), deﬁned as below: ∗ ) = ∧{∨cij∗ |1 ≤ j ≤ i ≤ |U|, cij = ∅} fD (a1∗ , . . 21) where cij∗ = {a ∗ |a ∈ cij }. The notation ∨{a, b, c, d} and ∧{a, b, c, d} denotes a ∨ b ∨ c ∨ d and a ∧ b ∧ c ∧ d, respectively. By ﬁnding the set of all prime implicants of the discernibility function, all the minimal reducts of a system may be determined. 2 the decision-relative discernibility function is (with duplicates removed) fD (a ∗ , b∗ , c∗ , d ∗ ) = (a ∗ ∨ b∗ ∨ c∗ ∨ d ∗ ) ∧ (a ∗ ∨ c∗ ∨ d ∗ ) ∧(b∗ ∨ c∗ ) ∧ (d ∗ ) ∧ (a ∗ ∨ b∗ ∨ c∗ ) ∧(a ∗ ∨ b∗ ∨ d ∗ ) ∧ (b∗ ∨ c∗ ∨ d ∗ ) ∧(a ∗ ∨ d ∗ ) Further simpliﬁcation can be performed by removing those clauses that are subsumed by others: fD (a ∗ , b∗ , c∗ , d ∗ ) = (b∗ ∨ c∗ ) ∧ (d ∗ ) The reducts of the dataset may be obtained by converting the expression above from conjunctive normal form to disjunctive normal form (without negations).

Am to a given entry of the discernibility matrix), deﬁned as below: ∗ ) = ∧{∨cij∗ |1 ≤ j ≤ i ≤ |U|, cij = ∅} fD (a1∗ , . . 21) where cij∗ = {a ∗ |a ∈ cij }. The notation ∨{a, b, c, d} and ∧{a, b, c, d} denotes a ∨ b ∨ c ∨ d and a ∧ b ∧ c ∧ d, respectively. By ﬁnding the set of all prime implicants of the discernibility function, all the minimal reducts of a system may be determined. 2 the decision-relative discernibility function is (with duplicates removed) fD (a ∗ , b∗ , c∗ , d ∗ ) = (a ∗ ∨ b∗ ∨ c∗ ∨ d ∗ ) ∧ (a ∗ ∨ c∗ ∨ d ∗ ) ∧(b∗ ∨ c∗ ) ∧ (d ∗ ) ∧ (a ∗ ∨ b∗ ∨ c∗ ) ∧(a ∗ ∨ b∗ ∨ d ∗ ) ∧ (b∗ ∨ c∗ ∨ d ∗ ) ∧(a ∗ ∨ d ∗ ) Further simpliﬁcation can be performed by removing those clauses that are subsumed by others: fD (a ∗ , b∗ , c∗ , d ∗ ) = (b∗ ∨ c∗ ) ∧ (d ∗ ) The reducts of the dataset may be obtained by converting the expression above from conjunctive normal form to disjunctive normal form (without negations).