Fuzzy sets

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Lecture 3

• Fuzzy sets

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1.1 Sets 1.1.1 Elements of sets An universal
set X is defined in the universe of discourse and
it includes all possible elements related with
the given problem. If we define a set A in the
universal set X, we see the following
relationships A ? X In this case, we say a set
A is included in the universal set X. If A is not
included in X, this relationship is represented
as follows. If an element x is included in the
set A, this element is called as a member of the
set and the following notation is used. x ? A
If the element x is not included in the set A,
we use the following notation. x ? A In
general, we represent a set by enumerating its
elements. For example, elements a1, a2, a3, , an
are the elements of set A, it is represented as
follows. A a1, a2, , an
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Another representing method of sets is given by
specifying the conditions of elements. For
example, if the elements of set B should satisfy
the conditions P1, P2, , Pn, then the set B is
defined by the following. B b b satisfies
p1, p 2, , pn In this case the symbol ..
implies the meaning of .such that.. In order to
represent the size of N-dimension Euclidean set,
the number of elements is used and this number is
called cardinality. The cardinality of set A is
denoted by A. If the cardinality A is a
finite number, the set A is a finite set. If A
is infinite, A is an infinite set. In general,
all the points in N-dimensional Euclidean vector
space are the elements of the universal set X.
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1.1.2 Relation between sets A set consists of
sets is called a family of sets. For example, a
family set containing sets A1, A2, is
represented by Ai i ? I where i is a set
identifier and I is an identification set. If all
the elements in set A are also elements of set B,
A is a subset of B. A ? B iff (if and only if) x
? A ? x ? B The symbol ? means .implication.. If
the following relation is satisfied,
A ? B and B ? A A and B have the same elements
and thus they are the same sets. This relation
is denoted by A B If the following relations
are satisfied between two sets A and B, A ? B
and A ? B then B has elements which is not
involved in A. In this case, A is called a
proper subset of B and this relation is denoted
by A ? B A set that has no element is called an
empty set Ø. An empty set can be a subset of any
set.
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1.1.3 Membership If we use membership function
(characteristic function or discrimination
function), we can represent whether an element x
is involved in a set A or not. Definition
(Membership function) For a set A, we define a
membership function µA such as µA(x) 1 if and
only if x ? A 0 if and only if x ? A We can say
that the function µA maps the elements in the
universal set X to the set 0,1. µA X ? 0,1
? As we know, the number of elements in a set A
is denoted by the cardinality A. A power set
P(A) is a family set containing the subsets of
set A. Therefore the number of elements in the
power set P(A) is represented by P(A) 2A
Example 1.1 If A a, b, c, then A 3 P(A)
Ø, a, b, a, b, a, c, b, c, a, b,
c P(A) 23 8 ?
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1.2 Operation of Sets 1.2.1 Complement The
relative complement set of set A to set B
consists of the elements which are in B but not
in A. The complement set can be defined by the
following formula. If the set B is the
universal set X, then this kind of complement is
an absolute complement set . That is, In
general, a complement set means the absolute
complement set. The complement set is always
involutive. The complement of an empty set is
the universal set, and vice versa.
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1.2.2 Union The union of sets A and B is defined
by the collection of whole elements of A and B.
A ? B x x ? A or x ? B The union might be
defined among multiple sets. For example, the
union of the sets in the following family can be
defined as follows. where the family of sets
is Ai i ? I The union of certain set A and
universal set X is reduced to the universal set.
A ? X X The union of certain set A and empty
set Ø is A. A ? Ø A The union of set A and
its complement set is the universal set.
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1.2.3 Intersection The intersection A n B
consists of whose elements are commonly included
in both sets A and B. A n B x x ? A and x ?
B The Intersection can be generalized between
the sets in a family of sets. where Ai i ?
I is a family of sets
The intersection between set A and universal set
X is A. A n X A The intersection of A and
empty set is empty set. A n Ø Ø The
intersection of A and its complement is all the
time empty set. When two sets A and B have
nothing in common, the relation is called as
disjoint. Namely, it is when the intersection of
A and B is empty set. A n B Ø
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1.2.4 Partition of Set Definition (Partition) A
decomposition of set A into disjoint subsets
whose union builds the set A is referred to a
partition. Suppose a partition of A is p, p(A)
Ai i ? I, Ai ? A then Ai satisfies following
three conditions. (1) Ø?iA(2) , ØnjiAAji, ?Iji?,
(3) ? AAIii?UIf there is no condition of (2),
p(A) becomes a cover or covering of the set A.
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1.4 Definition of fuzzy set 1.4.1 Expression for
fuzzy set Membership function µA in crisp set
maps whole members in universal set X to set
0,1. µA X ?0, 1 Definition (Membership
function of fuzzy set) In fuzzy sets, each
elements is mapped to 0,1 by membership
function. µA X ?0, 1 Where 0,1 means real
numbers between 0 and 1 (including 0,1).
Consequently, fuzzy set is vague boundary set
comparing with crisp set.
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Graphical representation of crisp and fuzzy sets
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Type-n Fuzzy Set
Example of type-2 fuzzy set
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Level-k fuzzy set
Example of level-2 fuzzy set
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