On the cardinality of fuzzy sets - EUSFLAT

On the cardinality of fuzzy sets
Pavol Kráˇl
Matej Bel University, Tajovského 40, 97401 Banská Bystrica
kral@fpv.umb.sk
Abstract
It seems that a suitably constructed fuzzy
sets of natural numbers form the most complete
and adequate description of cardinality
of finite fuzzy sets.(see [11]) Nevertheless,
in many applications one needs a
simple scalar evaluation of that cardinality
by nonnegative real number, e.g scalar
cardinality. There are many approaches to
this evaluation-sigma count of fuzzy set,psigma
count of fuzzy sets, cardinality of
its core or support, cardinality of its αcut
set,etc.(see [2]), [3], [4], [5], [7], [9],
[10] for more details). Wygralak in [8]
present an axiomatic approach to the scalar
cardinality of fuzzy sets which contains as
particular cases all standard concepts of
scalar cardinality. The best approximation
of scalar cardinality of fuzzy set is presented
in [12]. The aim of this paper is to select
numbers from Wygralak’s best approximation
which are coherent with human intuition.
This selection is based on the dependency
among objects which can be represented
by fuzzy compatibility.
Keywords: cardinality of fuzzy sets, dependency
among objects, fuzzy compatibility.
1 Introduction
Let X be an universal set. Fuzzy sets in X will be identified
with their membership functions A : X → [0,1].
The cardinality of fuzzy set A will be denoted by
card(A). The cardinality of crisp set B will be denoted
by |B|. Fuzzy compatibility is a function C :
X 2 → [0;1] fulfilling the following properties
C(a,a) = 1,
C(a,b) = C(b,a),
for each a,b ∈ X. We use the following basic definitions
of fuzzy set theory:
Aα = {x ∈ X : A(x) ≥ α}
with α ∈]0,1] (α-cut set of A),
A α = {x ∈ X : A(x) > α}
with α ∈ [0,1[ (sharp α-cut set of A ),
(core of A),
core(A) = Aα
supp(A) = A 0
(support of A ). If supp(A) is finite, we shall say that
A is a finite fuzzy set. Measuring the cardinality of a
finite fuzzy set has many applications. (fuzzy querying
in databases, expert systems, evaluation of natural
language statements, aggregation, decision making in
fuzzy environment, metrical analysis of grey images,
etc. (see [3], [7], [15] for more details and further reference).
One of the scalar cardinalities is the cardinality
described by Wygralak in [12]. Wygralak proposes
one of the values of the interval
[|A 0.5 |,|A0.5|],
as the best scalar approximation of cardinality of
fuzzy set. Some authors think that only valid values
for the cardinality of fuzzy set are the cardinalities
of its α-cuts because only this representation preserve
the dependency among objects from the point of view
of cardinality.

Example 1.1 [1] Let X={John, Mike, Peter} be a set
of friends, and let us suppose that John is blonde, and
Mike and Peter are fairly blonde. We could say that
John is in the set of blonde people with degree 1 and
Mike and Peter are in the same set with degree 0.5.
Suppose we want to buy a cap for every blonde person
in X. Deciding how many caps we shall buy is
equivalent to obtaining the cardinality of the fuzzy set
XBLONDE = 1/John + 0.5/Mike + 0.5/Peter.
Following Wygralak’s cardinality, the possible cardinalities
for XBLONDE are [1,3]. If we are strict with
the concept of blonde, we shall buy only one cap (for
John). But if we relax our criterion we could think of
buying three caps. It is clear that buying two caps
is not a solution for the problem, i.e cardinality of
XBLONDE can be one or three, but not two. (Mike is
in XBLONDE iff Peter also is )
In our opinion, only for α “near”1 is valid to take into
account the dependency among objects with respect
to the property “to be in A ”. In other cases the fuzzy
compatibility is more suitable to express dependency
among objects. Let us consider the following example:
Example 1.2 Let A = 0.5/x1 + 0.5/x2 + 0.5/x3 +
0.5/x4 be a fuzzy set. The possible cardinality for A
belong to [0,4]. If we assume the dependency among
objects (from the point of view of cardinality) the possible
cardinalities are 4 and 0, i.e. each xi is in A
or none of xi is in A. Following properties of an entropy
measure, proposed in [2], A is the fuzziest set
over four objects, i.e we can assume each integer from
[0,4] as a possible cardinality of A (integer from [0,4]
are equivalent) and we need additional property of objects
xi to select the best approximation of cardinality.
The natural additional property of objects xi is a fuzzy
compatibility for example.
We will define the selection from Wygralak’s best approximation
which is more appropriate.
Definition 1.1 Let A be a finite fuzzy subset of X.
Let A(x1),A(x2),...A(xn) are the values of A. Let
C : A 2 → [0,1] be a fuzzy compatibility. Let 0.5 A =
{x ∈ X : A(x) = 0.5}. The possible scalar cardinality
of fuzzy set A with respect to the fuzzy compatibility
is:
D + A
card(A) =
0.5 i f D > 1
A0.5 else
where D = max
x∈0.5
|{y ∈0.5 A,: C(x,y) ≥ β : |} and
A
β ∈ [0,1] is a treshold.
Example 1.3 Let A from example 1.2 be a set of middle
aged people. Let x1, x2 are 20 year old and x3,
x4 are 60 year old. Let β = 1. Let C1(xi,xj) = 1 iff
xi,xj are in the same age and C1(xi,xj) = 0 else. It
is easy to see that card(A) = 2. Let C2(xi,xj) = 1 iff
A(xi) = A(x j) and C2(xi,xj) = 0 else. It is easy to see
that card(A) = 4.
Proposition 1.1 Let A be a finite fuzzy set. Let C :
A 2 → [0,1] be a fixed fuzzy compatibility.
1. If β → 0 card(A) → |A0.5|
2. If β → 1 card(A) → |A 0.5 |
Proposition 1.2 Let A be a finite fuzzy set. Let β be a
fixed treshold
1. If for all i, j,i = j C(xi,xj) < β card(A) = |A 0.5 |
2. If for all i, j,i = j C(xi,xj) ≥ β card(A) = |A0.5|
Proposition 1.3 Let C be a fixed compatibility. Let β
be a fixed treshold. We can construct a naive cardinality
theory.
The selection of a compatibility and a treshold depends
on a concrete case.
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