Applied Mathematical Sciences, Vol. 8, 2014, no. 41, 2035 - 2040
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.42104
Near Sets through Fuzzy Similarity Relation
Rogi Jacob
Department of Mathematics, U.C. College, Aluva
Mahatma Gandhi University
Kerala, India
Sunny Kuriakose A
Principal, B. P. C. College, Piravom, 686 664, India
Copyright © 2014 Rogi Jacob and Sunny Kuriakose A. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Abstract
Near sets are considered as an extension of rough sets. They are disjoint sets that resemble
each other. In this paper, we proposed a class of near sets formed through a fuzzy binary
relation and its partition. The application of near fuzzy set in the field of image segmentation
evaluation is also mentioned.
Keywords: Near sets, Fuzzy sets, Fuzzy similarity relation
1. Introduction
Fuzzy sets provide a natural frame work for dealing with uncertainty. It offers a problem
solving tool between the precision of classical mathematics and the inherent precision of the
real world. It already established the importance in different application of real life. On the
other hand Near set theory introduced by J.F.Peters[1][2]is fairly a new intelligent technique
for managing uncertainty that is used for the discovery of data dependencies, to evaluate the
importance of attributes, to discover patterns in data and to recognize and classify objects.
Rogi Jacob and Sunny Kuriakose A
2036
Most of these paradigms are dedicated to approximately solving real world problems in
pattern classification and learning [5] [6][9].
Near sets are considered as an extension of rough sets[4][7]. They are disjoint sets that
resemble each other. Resemblance between disjoint sets occurs whenever there are
observable similarities between the objects in the sets. Similarity is determined by comparing
lists of object feature values. Each list of feature values defines an object's description.
Comparison of object descriptions provides a basis for determining the extent that disjoint
sets resemble each other. Objects that are perceived as similar based on their descriptions are
grouped together. These groups of similar objects can provide information and reveal patterns
about objects of interest in the disjoint sets. In this paper, we attempted to join the near set
paradigm with fuzzy set to obtain better results in the classification scenario.
2. Preliminaries
In this section, we recall certain standard definitions already in the literature. Throughout this
paper, we use the fuzzy binary relation ‘ γ ’ given in [8]. Each value of α ∈ Λ(γ ) induces a
fuzzy similarity relation, ‘ α γ ’. An overview of symbols is given in Table 1.
Butin near set, the elements are perceptual objects. The discovery of near sets begins with
choosing the appropriate method to describe observed objects .These observed object may be
anything existing in the real world. It provides a formal basis for the observation, comparison
and classification of objects. The common terminologies used in near set theory are given in
the following definitions:
Definition1 [2]: A perceptual object is something perceivable that has its origin in the
physical world.
Definition 2 [3]:
perceptual object.
Definition 3[1]:
A probe function is a real-valued function representing a feature of
A perceptual system O, F
consists of a nonempty set O of sample
perceptual objects and a nonempty set F of real-valued functions φ ∈ F such that φ :O → R
Near sets through fuzzy similarity relation
2037
Table 1
Symbols Description of symbols
γ
Fuzzy similarity relation
α
Alpha cut of γ
γ
Λ(γ )
Level set
O
Set of perceptual objects
O /αi
Quotient set
[ x ]α
Equivalence class
i
∼
indiscernibility relation
weak indiscernibility relation
(ϖ )
Weakly nearness fuzzy relation
ω
nearness fuzzy relation
3. Near sets
Let X is any universal set with finite number of elements and γ be the fuzzy similarity
relation defined on X. For each value of α ∈Λ(γ ) , α γ is a fuzzy equivalence relation and it
splits X into disjoint equivalence class. Let ‘ O ’ denote the set of all disjoint equivalence
classes obtained by each α i . In other words for each α i , we got a fuzzy partition O /αi . Then
naturally a question arises, which partition will yield a better result? Here we attempt to
answer this question by the following definitions.
Definition 1: A Fuzzy perceptual system O, F
consists of a nonempty set O of sample
perceptual objects and a nonempty set ℑ of fuzzy- membership functions φ ∈ ℑ such that
φ : O → [0,1]
Rogi Jacob and Sunny Kuriakose A
2038
Definition 2: Let ( Ο, ℑ) be a fuzzy perceptual system. ∀ B ⊆ ℑ , the indiscernibility relation
is defined as follows:
}
∼ B = {([ x ]α , [ y ]α ) ∈ O × O / ∀Φ / ~ B such that O / α ( x ) = O / α ( y )
i
j
i
j
Definition 3: Let ( Ο, ℑ) be a fuzzy perceptual system. ∀ B ⊆ ℑ , the weak indiscernibility
relation is defined as follows:
}
B = {([ x ]α , [ y ]α ) ∈ O × O / ∃O / ~ B ∈ B such that O / α ( x ) = O / α ( y )
i
j
i
j
Definition 3(Weak near fuzzy relation): Let ( Ο, ℑ) be a fuzzy perceptual system and
X ,Y ⊆ O . A set X is weakly near fuzzy (ϖ ) to set Y within the fuzzy perceptual system
( Ο, ℑ) ⇔ ∃ an x ∈ X and
y ∈Y such that x B y for some B ⊆ ℑ
Definition 4 (Near Fuzzy relation): Let ( Ο, ℑ) be a fuzzy perceptual system and X ,Y ⊆ O . A
set X is near fuzzy (ω ) to set Y within the fuzzy perceptual system
( Ο, ℑ)
⇔ ∀ x ∈ X and y ∈Y such that x ∼ ℑ y
Definition 5 (Near fuzzy set): A set X is near fuzzy to a set Y ⇔ x ω y ∀ x ∈ X and y ∈ Y
Definition 6 (Weakly near fuzzy set): A set X is weakly near fuzzy to a set Y
⇔ ∃ an x ∈ X and y ∈ Y such that xϖ y
Table 2
Consider an example to illustrate the above definitions:
12 10
Let X represents a digital image as given in Table 2.
0
14
Then fuzzy relation [ ] obtained from above pixel values isexpressed as a relation
41 20
matrix,Q,
125 0


Q
1

.21
.29

1
.21
= .09

.12
 .09

1
.12


.21
1
.29
.13
1 .21
1 .15
.09
.29
.13
1
1
.08
.09
1
.15
1
.08
1
1
1
1
1
.08
.29
.29
.09
.06
1
1
.08
.07
1
.02
.55
1
.11
1
1
1
.10
1
.27
1
.13 .52
1 .07
.02
.12
.29
.09 1 .12

.55 1 .13 
.06 .11 1 .52 

1
1 1
1 
.07 .10 1 .07 

.02 .27
1 .02 

1 .27 1 .18 
.27 1 1 .12 

1 1 1 1 
.18 .12 1 1 


Obviously Q is a fuzzy similarity relation and the level set of Q is given by
Λ (Q) = {0.02,0.06,0.07,0.08,0.09,.10,.11,.12,.13,.15,.18,.21,.27,.55,1}. For each α ε Λ(Q),
define an equivalence relation α Q and the corresponding equivalence relation partition the
given set X into disjoint equivalent classes . The partition corresponding to some values of
25
81
0
0
Near sets through fuzzy similarity relation
2039
alpha are given in figure1. The same colours represent the matching descriptions of elements
in the given set X. Hence the weak near set obtained using the partition is denoted by
Nearϖ (O ) and is given by Nearϖ (O ) = {( 2;5) , ( 4;9 ) , ( 3,5)( 6,8 ) ( 7,10 )}
Proposition: Let O, F be a fuzzy perceptual system andlet X , Y ⊆ O . Then the following
conditions are equivalent.
1. X ω Y .
2. there are x ∈ X , y ∈ Y and thereis B ∈ F such that x ∼ B y
3. there are x ∈ X , y ∈ Y and thereis B ∈ F such that x B y
proof : from definitions
4. Conclusion
This paper has proposed a new class of near sets formed from the fuzzy similarity relation. It
is shown (with example) that near fuzzy set can be applicable to the evaluation of
classification of fuzzy sets. The further work should be done to explore the theoretical
features as well as practical implications of the introduced near fuzzy sets.
References
[1] James. F. Peters and P. Wasilewski, Foundations of near sets, Information sciences, 179
(2009), 3091-3109.
2040
Rogi Jacob and Sunny Kuriakose A
[2] Christopher. J. Henry, Neighbourhoods, Classes and Near sets, Applied mathematical
sciences, 5-35 (2011), 1727-1732.
[3] James. F. Peters, Near sets. General theory about nearness of objects, 1-53 (2007), 26092629
[4] N.Senthilkumaran and R.Rajesh, A study on rough set theory for medical image
segmentation, International journal of recent trends in Engineering, 2-2 (2009), 236-238.
[5] Jim peters and SomNaimpally, Applications of near sets, Notices of the AMS, 59-4
(2012), 536-542.
[6] Christopher Henry and James. F. Peters, Near set evaluation and recognition (Near
System), UM CI Laboratory technical report, 2012.
[7] Aboul Ella Hassanien, Ajith Abraham, James. F. Peters and Gerald Schaefer, Rough sets
and near sets in medical imaging: a review, IEEE Trans. on information technology in
biomedicine, 10-10 (2008), 1-14.
[8] Rogi Jacob, Sunny Kuriakose A and Sony George, Medical Image segmentation using
fuzzy binary relation, Int. journal of fuzzy mathematics and systems, 2-1 (2012), 1-10.
[9] ZdziskawPawlak, Rough Classification, Int.J.of human-computer studies, 51(1999), 369283.
Received: February 17, 2014