Structure-Based Attribute Reduction: A Rough Set Approach - Feature Selection for Data and Pattern Recognition - page 151

Table 7.8 The decision table
in Table 7.7 with the
generalized decision function

Student Ma Ph Li E ʴ = l C , u C C

u 1 Good Good Good Good good, good

u 2 Good Good Med Med med, good

u 3 Med Good Med Good med, good

u 4 Bad Med Good Med med, med

u 5 Med Bad Med Bad bad, med

u 6 Med Bad Bad Med bad, med

u 7 Bad Bad Bad Bad bad, bad

Non-domination matrices M are obtained as follows.
u 1 u 2 u 3 u 4 u 5 u 6 u 7

u 1 ∅{

}{ , }{ , }

Li Ma Li Ma Ph C C C

u 2 ∅∅ {

}{ , }{ , }

Ma Ma Ph Ma Ph C C

u 3 ∅∅ ∅ { Ma , Ph }{ Ph }{ Ph , Li } C

u 4 ∅{ Li }{ Li } ∅ { Ph , Li }{ Ph , Li }{ Ph , Li }

u 5 ∅∅ ∅ { Ma } ∅ { Li }{ Ma , Li }

u 6 ∅∅ ∅ { Ma } ∅ ∅ { Ma }

u 7 ∅∅ ∅

∅ ∅ ∅ ∅

For example, the entry corresponding to row u 1 and column u 3 on M contains
Ma and Li, because u 3 is worse than u 1 with respect to Ma and Li but not worse
with respect to Ph. Symbol C at some entries means {Ma, Ph, Li}. The rows with
symbol ∗ show objects u i such that l C ( u i ) = u C ( u i ) .

The Boolean function F ≥ is obtained from M as

F ≥ ( Ma , Ph , Li = Ph ∧ Li ) =

m ij ˜

c ∧

m ij ˜

c .

c ∈ c ∈

i = 1 , j = 2 , 3 ,..., 7 i = 2 , 3 , 4 , j = 5 , 6 , 7

From the last equation, F ≥ ( Ma , Ph , Li true only when Ph true and Li true.

This implies that only {Ma, Ph, Li} and {Ph, Li} satisfy (DL ≥ ) owing to Theorem 10 .
An L ≥ -reduct is a minimal set of condition attributes that satisfies (DL ≥ ). Therefore,
{Ph, Li} is a unique L ≥ -reduct. Moreover, the L ≥ -reduct corresponds to a unique
prime implicant of F ≥ , i.e., Ph

) = = =

∧ Li.
Similarly, Boolean functions F ≤ , F U and F L are

F ≤ (

Ma Ph Li Ma Ph

, , ) = c ∧ c = ∧ ,

c ∈ m ji c ∈ m ji

i = 4 , 5 , 6 , 7 , j = 1 , 2 , 3 i = 7 , j = 1 , 2 ,..., 6

F L

Ma Ph Li

( , , ) = c ∧ c ∧ c

c ∈ m ij c ∈ m ij c ∈ m ji

i = 1 , j = 2 , 3 ,..., 7 i = 4 , j = 5 , 6 , 7 i = 4 , j = 1 , 2 , 3

Ma Li

∧ c = ∧ .

c ∈ m ji

i = 7 j = 1 , 2 ,..., 6