International Journal of Engineering and Advanced Technology (IJEAT)
ISSN: 2249 – 8958, Volume-9, Issue-1S4, December 2019
397
Retrieval Number: A11931291S419/2019©BEIESP
DOI: 10.35940/ijeat.A1193.1291S419
Published By:
Blue Eyes Intelligence Engineering
& Sciences Publication
Combinations of Fuzzy Rough Covers and
Covering Principles
D. Vidhya
Abstract-Rough set deals the mathematical approach to
uncertain knowledge. It has many interesting applications in the
area of machine learning, decision analysis, pattern recognition
etc., A fuzzy rough set is a generalization of a rough set, derived
from the approximation of a fuzzy set. The focus of fuzzy rough
set is to define lower and upper approximation from the universal
set. This paper gives four operations of open covers and three
notions of covering principles in fuzzy rough set. Also
investigates the relations of covers and principles. Finally
combine the open covers and principles, find the relationship
between them.
Keywords— Fuzzy rough cover(s), covering principle(s)
I. INTRODUCTION
Authors [1], [6] and [7] said the concepts and
properties of fuzzy sets, applications of fuzzy set and fuzzy
topological spaces. Further various uncertainities that arise
in the real world problems are solved by using rough set
theory [3, 5], intuitionistic fuzzy rough set theory [2] etc. S.
Nanda and S. Majumdar [4] analysed the concept of fuzzy
rough set. This paper introduces the concepts of types of
fuzzy rough open cover and types of fuzzy rough covering
principle. Some basic properties are also established. In
particularly, the concept of relationship between them are
discussed.
II. PRELIMINARIES
Definition 2.1 [4]
=
,
and =
,
in ,
1) F=B if and only if μ
F
L
a = μ
B
L
a for each
a ∈ Z
L
and μ
F
U
a = μ
B
U
a for each a ∈ Z
U
.
2) F ⊆ B if and only if μ
F
L
e≤μ
B
L
efor each
e ∈ Z
L
and μ
F
U
e≤μ
B
U
efor every e ∈ Z
U
.
3) C=F ∪ B if and only if
μ
C
L
r = max[μ
F
L
r, μ
B
L
r] for all r ∈ Z
L
and
μ
C
L
r = max[μ
F
L
r, μ
B
r] for all r ∈ Z
U
.
4) D=F ∩ B if and only if
μ
D
L
f = min[μ
F
L
f, μ
B
L
f] for all f ∈ Z
L
and
μ
D
U
f = min [μ
F
f, μ
U
f] for all f ∈ Z
U
.
A complete lattice is for any indexed set ,
: ∈ is
(), =∪
if and only if
=
∈
() for
all ∈
and
=
∈
() for all ∈
.
Similarly, =∩
iff
=
∈
() for all
∈
and
=
∈
() for all ∈
for all
∈
.
Revised Manuscript Received on December 15, 2019.
D. Vidhya, Department of Mathematics, Vels Institute of Science,
Technology and Advanced Studies, Pallavaram, Chennai, India. E-mail:
vidhya.d85@gmail.com
A complement ′ of is defined by
′
,
′ of
=
1 −
for all ∈
and
=1 −
for all
∈
.
Defintion 2.2 [8]
A fuzzy rough set =
,
in Z is defined by
:
→ and
:
→ with
() ≤
() for every
∈
. The group of all fuzzy rough sets in Zis().
Definition 2.3 [8]
Any fuzzy rough topology on a rough set X is a
family T of fuzzy rough sets in X which satisfies the
following conditions:
1) 0
,1
∈
2) If , ∈ then ∩∈ and
3) If
∈ for all ∈, then
∪
∈
∈.
Therefore X together with topology T is called a fuzzy rough
topological space. The members of T are open and its
complement is closed.
Defintion 2.4 [8]
=
,
is called neighbourhood of =
,
iff there exists fuzzy rough open set =
,
such that ⊆⊆.
III. FUZZY ROUGH OPEN COVERS SEQUENCE
AND COVERING SEQUENCE PRINCIPLES
This section introduces the types of rough open
cover sequences and covering principles and discuss their
characterizations and relationship between them.
Definition 3.1
A collection Σ =
: ∈ of rough sets in X is fuzzy
rough cover of 1
if ∪
∈
=1
. Each members of Σ are open
iff a collection Σ is fuzzy rough open cover.
Definition 3.2
Each open cover has a finite subcover then the space is
compact.
Defintion 3.3
If
= ,
,
: , ∈ ℕ for ∈
,
∈ [0,1] and = then
∈ℕ
is a fuzzy
rough sequence.
A. Fuzzy Rough Open Covers
Definition 3.1.1
A covering sequence of a rough set =
,
is a
collection Θ =
∈ℕ
∈
if
∪
∈
∈ℕ
= .