SLIDE 6 21
B-positive region of A
Consistent/Inconsistent Decision Systems
U x x tic determinis consistent i x d x A IND x U x i x V U
A A d A
∈ = ∂ = ∈ ∃ = ∂ → ∂ any for 1 | ) ( | if ) ( is } ) ( and ) ( ' ' | { ) ( ) ( : A P
Example Walk is inconsistent, Hiring is consistent Corollary
A B B d POS d POS U d POS
B B B B A
⊆ = ∂ = ∂ = ' , sets empty
pair any for ) ( ) ( then , If consistent is iff ) (
' '
A
22
B-positive region of A
). ( by denoted is and the called is ... set then the ,
classes decision are If
) ( 1 ) ( 1
d POS
region positive
X B X B , X , X
B d r d r
A ∪ ∪ … A
A A
Example U X A X A Hiring U X A X A Walk
Reject Accept No Yes
= ∪ ≠ ∪ : ble decison ta : able decision t
23
The Classification Problem
Example Let A=(U, A∪{d}) be consistent, M(A) = (cij) its discernibility matrix. Construct the decision-relative discernibility matrix of A: Md(A) = (cd
ij) assuming cd ij = ∅ if d(xi) = d(xj), and
cd
ij = cij - {d} otherwise.
Each of the two decision-relative reducts: {Experience, Reference} and {Diploma, Experience} uniquely defines to which decision class an object belongs
Accept Neutral Yes High MSc x4 Reject Neutral Yes Medium MSc x5 No Yes Yes No Yes Yes French Low Low Low High High Medium Experience Excellent Good Neutral Good Excellent Excellent Reference Reject MCE x8 Reject MCE x3 Reject MBA x2 Accept MBA x7 Accept MSc x6 Accept MBA x1 Decision Diploma
24
Discerning Objects from Different Classes
d, e, r d, e, f, r d, e, f e, f, r ∅ [x7] d, e, f e, r d, e, r d, e, r ∅ ∅ [x6] d, e, f, r e d, e, r d, e ∅ ∅ ∅ [x4] d, e, f d, r d, e, r e, r ∅ ∅ ∅ ∅ [x1] ∅ ∅ ∅ [x3] ∅ ∅ ∅ ∅ [x2] ∅ ∅ [x5] ∅ [x8] [x5] [x8] [x3] [x2] [x7] [x6] [x4] [x1]
Example Hiring: the decision-relative discernibility matrix Reduced decision-relative discernibility function: fdM (A) = ed ∨ er
