The document discusses fuzzy sets and rough sets. It explains that fuzzy sets allow partial membership, where an element can belong to a set to a degree between 0 and 1, while in classical sets an element either fully belongs or does not belong. Rough sets address vagueness through boundary regions rather than partial membership. Rough sets are defined using topological operations on lower and upper approximations. Indiscernibility relations are also discussed, where indiscernible elements cannot be distinguished based on available attributes.
2. FuzzyFuzzy SetsSets
• Dr. Lotfi Zadeh propose this approach.
• In his approach an element can belong to a
degree k (0 <= k <= 1).
• In classical set theory an element must belong or
not belong to a set.
• Fuzzy membership function can be presented as:
µX(x) є (0,1)
where, X is a set and x is an element.
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3. Fuzzy Sets (Continue)Fuzzy Sets (Continue)
• Fuzzy membership function has the following
properties.
• µU -x(x) = 1 - µX(x) for any x є U
• µxUy(x) = max(µX(x), µy(x)) for any x є U
• µx ∩y(x) = min(µX(x), µy(x)) for any x є U
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4. Rough SetsRough Sets
• Rough set theory is another approach to handle
vagueness.
• Imprecision in this approach is expressed by a
boundary region of a set, and not by a partial
membership, like in fuzzy set theory.
• Rough set concept can be defined by topological
operation interior and closure called
approximations.
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5. Rough Sets (Continue)Rough Sets (Continue)
• Suppose we are given a set of objects U called
the universe and an indiscernibility relation R ⊆ U
× U, representing our lack of knowledge about
elements of U.
• For simplicity we assume that R is an equivalence
relation.
• Let X be a subset of U.
• We want to characterize the set X with respect to
R. To do this we will need the basic concepts of
rough set theory given in next slide. 5
6. Rough Sets (Continue)Rough Sets (Continue)
• The lower approximation of a set X with respect
to R is the set of all objects, which can be for
certain(sure) classified as X with respect to R (are
certainly X with respect to R).
• The upper approximation of a set X with respect
to R is the set of all objects which can be
possibly(maybe) classified as X with respect to R
(are possibly X in view of R).
• The boundary region of a set X with respect to R is
the set of all objects, which can be classified
neither as X nor as not-X with respect to R. 6
7. Rough Sets (Continue)Rough Sets (Continue)
• So that,
• Set X is crisp (Exact with respect to R), if the
boundary region of X is empty.
• Set X is rough (Inexact with respect to R), if the
boundary region of X is nonempty.
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8. Rough ApproximationRough Approximation
• Formal definitions of approximations and the
boundary region are as follows:
• R-lower approximation of X
R*(x) = U {R(x): R(x) ⊆ X}
• R- upper approximation of X
R*
(x) = U {R(x): R(x) ∩ X ≠ ɸ}
• R-boundary approximation of X
RNR (X) = R*
(X) - R*
(X)
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9. Rough ApproximationRough Approximation
• As we can see from the definition, approximations
are expressed in terms of granules of knowledge.
• The lower approximation of a set is union of all
granules which are entirely included in the set.
• The upper approximation − is union of all granules
which have non-empty intersection with the set.
• The boundary region of set is the difference
between the upper and the lower approximation.
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10. Rough Membership functionRough Membership function
• Rough sets can be also defined as rough
membership function.
µx
R
: U (0,1)
Where
µx
R
(x) = |X∩ R(x)| / |R(x)|
And |X| denotes the cardinality of X.
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11. Rough MembershipRough Membership
function(Conti)function(Conti)
• The rough membership function can be expresses
conditional probability.
• That x belongs to X given R.
• And can be interpreted as a degree that x belongs
to X in view of information about x expressed by
R.
• The meaning of rough membership function can
be defined as shown in fig 1.
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13. Rough MembershipRough Membership
function(Conti)function(Conti)
• The rough membership function can be used to
define approximations and the boundary region
of a set, as shown below:
R*
(x) = {xєU : µx
R
(x) = 1}
R*
(x) = {xєU : µx
R
(x) > 0}
RNR (X) = {xєU : 0 < µx
R
(x) < 1 }
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14. Rough MembershipRough Membership
function(Conti)function(Conti)
• The rough membership function has the
following properties:
• µx
R
(x) = 1 iff x є R*
(x)
• µx
R
(x) = 0 iff x є U - R*
(x)
• 0 < µx
R
(x) < 1 iff x є RNR (X)
• µR
U-x (x) = 1 - µx
R
(x) for any x є U
• µxUy(x) => max(µR
X(x), µR
y(x)) for any x є U
• µx ∩y(x) <= min (µR
X(x), µR
y(x)) for any x є U
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15. IndiscernibilityIndiscernibility
• A decision system (i.e. a decision table) express
all the model.
• This table may be unnecessarily large.
• The same or indiscernible objects may be
represented several times.
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16. Indiscernibility (Conti)Indiscernibility (Conti)
Element Age LEMS Walk
X1 16-30 50 Yes
X2 16-30 0 No
X3 31-45 1-25 No
X4 31-45 1-25 Yes
X5 46-60 26-49 No
X6 16-30 26-49 No
X7 46-60 26-49 Yes
LEMS = Lower Extremity(boundary) Motor Score
Table 1
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17. IndiscernibilityIndiscernibility
• A binary relation R ⊆ X x X which is reflexive (i.e.
an object is in relation with itself xRx), symmetric
(if xRy then yRx) and transitive (if xRy and yRz
then xRz) is called an equivalence relation.
• The equivalence class of an element x є X
consists of all objects y є X such that xRy.
• Let A = (U, A) be an information system then with
any B ⊆ A there is associated an equivalence
relation INDA (B) :
INDA (B) = {(x,x’) є U2
| a є B a(x) = a(x’)} 17
18. IndiscernibilityIndiscernibility
• INDA (B) is called the B-indiscernibility relation. If
(x,x') є INDA (B) then x and x' are indiscernible
from each other by attributes from B.
• The equivalence classes of the B-indiscernibility
relation are denoted [x]B.
Example:
• Let us illustrate how a decision table such as
Table 1 defines an indiscernibility relation.
• The non-empty subsets of the conditional
attributes are {Age}, {LEMS} and {Age, LEMS}. 18
19. IndiscernibilityIndiscernibility
• If we consider, for instance, {LEMS}, objects X3 and
X4 belong to the same equivalence class and are
indiscernible.
• The relation IND defines three partitions of the
universe.
IND ({Age}) = {{x1,x2,x6},{x3,x4},{x5,x7}}
IND ({LEMS}) = {{x1},{x2},{x6},{x3,x4},{x5,x6,x7}}
IND ({Age,LEMS}) = {{x1},{x2},x6},{x3,x4},{x5,x7},{x6}}
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