Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
21,
No.
2,
February
2021,
pp.
1030
1038
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v21i2.pp1030-1038
r
1030
Pythagor
ean
Fuzzy
N
-Soft
Gr
oups
M.
Shazib
Hameed
1
,
Salman
Mukhtar
2
,
Haq
Nawaz
Khan
3
,
Shahbaz
Ali
4
,
M.
Haris
Mateen
5
,
Muhammad
Gulzar
6
1,2,3,4
Department
of
Mathematics,
Kha
w
aja
F
areed
Uni
v
ersity
of
Engineering
and
Information
T
echnology
,
64200,
Rahim
Y
ar
Kahn,
P
akistan
5
Department
of
Mathematics,
Uni
v
ersity
of
Punjab
Lahore,
54590,P
akistan
6
Department
of
Mathematics,
Go
v
ernment
Colle
ge
Uni
v
ersity
F
aisalabad,
38000,
P
akistan
Article
Inf
o
Article
history:
Recei
v
ed
Jul
18,
2020
Re
vised
Aug
11,
2020
Accepted
Sep
7,
2020
K
eyw
ords:
(
;
;
)
-cut
subgroups,
(
;
;
)
-cut
of
Pythagorean
fuzzy
N
-soft
set,
Pythagorean
fuzzy
N
-soft
groups,
Pythagorean
fuzzy
N
-soft
sets
complements,
Pythagorean
fuzzy
N
-soft
sets
ABSTRA
CT
W
e
elaborate
in
this
paper
a
ne
w
structure
Pythagorean
fuzzy
N
-soft
groups
which
is
the
generalization
of
intuitionistic
fuzzy
soft
group
initiated
by
Karaaslan
in
2013.
In
Pythagorean
fuzzy
N
-soft
sets
concepts
of
fuzzy
sets,
soft
sets,
N
-soft
sets,
fuzzy
soft
sets,
intuitionistic
fuzzy
sets,
intuiti
onistic
fuzzy
soft
sets,
Pythagorean
fuzzy
sets,
Pythagorean
fuzzy
soft
se
ts
are
generali
zed.
W
e
also
talk
about
some
elementary
ba-
sic
concepts
and
operations
on
Pythagorean
fuzzy
N
-soft
sets
with
the
assistanc
e
of
illusions.
W
e
additionall
y
define
three
dif
ferent
sorts
of
complements
for
Pythagorean
fuzzy
N
-soft
sets
and
e
xamined
a
fe
w
outcomes
not
hold
in
Pythagorean
fuzzy
N-soft
sets
complements
as
the
y
hold
in
crisp
set
h
ypothesis
with
the
assistance
of
counter
e
xamples.
W
e
further
talk
ed
about
(
;
;
)
-cut
of
Pythagorean
fuzzy
N
-soft
set
and
their
properties.
W
e
lik
e
wise
talk
about
some
essential
properties
of
Pythagorean
fuzzy
N
-soft
groups
lik
e
groupoid,
normal
group,
left
and
right
cosets,
(
;
;
)
-cut
subgroups
and
some
fundamental
outcom
es
identified
with
these
terms.
Pythagorean
fuzzy
N
-soft
sets
is
increasingly
ef
ficient
and
adaptable
model
to
manage
uncertain-
ties.
The
proposed
models
of
Pythagorean
fuzzy
N
-soft
groups
can
defeat
a
fe
w
dis-
adv
antages
of
the
e
xisting
statures.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
M.
Haris
Mateen
Department
of
mathematics
Uni
v
ersity
of
Punjab
Lahore
54590,P
akistan
Email:
harism.math@gmail.com
1.
INTR
ODUCTION
T
o
handel
comple
x
genuine
issues
the
techniques
generally
utilized
in
traditional
mathematics
are
not
constantly
beneficial
because
of
a
number
of
sorts
of
v
agueness
and
uncertainties
in
v
olv
e
in
such
issues.
There
are
v
arious
procedures,
similar
to
the
interv
al
mathematics,
h
ypothesis
of
probability
,
rough
set
h
ypothesis,
and
fuzzy
set
h
ypothesis
which
we
can
be
consider
of
as
numerical
models
for
adapting
to
vulnerabilities.
Unfor
-
tunately
,
e
v
ery
one
of
these
structures
ha
v
e
their
o
wn
do
wn
sides
and
troubles.
F
or
instance,
in
the
genuine
the
w
ords
lo
v
ely
,
best,
reno
wned
are
ambiguous.
The
criteria
for
w
onderful,
best,
reno
wned
fluctuates
from
indi
vidual
to
indi
vidual.
T
o
handel
such
type
of
ambiguous
and
uncertain
information
initiated
the
the
possi-
bility
of
fuzzy
set
augmentation
of
the
customary
ne
w
sort
with
the
assistance
of
participation
function
which
assign
an
enrollment
grade
in
the
unit
shut
interv
al
[0
;
1]
to
e
v
ery
alternati
v
e
in
the
uni
v
erse
[1].
The
possibility
of
intuitionis
tic
fuzzy
sets
as
an
generalization
of
fuzzy
set
by
introducing
the
idea
of
enrollment
and
non-
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1031
participation
grades
[2,
3].
The
idea
of
a
latest
sort
or
ne
w
sort
of
set
also
called
soft
set,
as
a
numerical
model
to
manage
uncertainties.
The
possibility
of
soft
s
ets
ha
v
e
opened
ne
w
entryw
ays
for
the
scientists
because
of
their
appl
ications
in
dif
ferent
genuine
issues.
Fuzzy
sets
and
soft
sets
are
independent
generalizations
of
crisp
sets
[4].
Pythagorean
fuzzy
set
introduced
in
[5].
In
[6–8],
certain
outcomes
for
Pythagorean
fuzzy
sets
and
some
Pythagorean
fuzzy
information
measures
and
their
implementations
in
v
estig
ated.
Soft
sets
gi
v
e
binary
object
e
v
aluation
and
other
numerics
models
lik
e
fuzzy
sets,
intuitionistic
fuzzy
sets
and
neutrosophic
sets
partner
esteems
in
the
interv
al
[0
;
1]
.
These
models
f
ail
to
handle
the
situation
while
modelling
partner
non-binary
decisions
on
true
issues.
In
addition
,
non-binary
asses
sments
in
the
rating
or
ranking
positions
are
e
xpected.
The
ranking
can
be
communicated
in
multinary
v
alues
in
the
type
of
number
of
stars,
spots,
grades
or
an
y
generalized
notation.
Propelled
by
these
concerns,
in
[9],
initiated
the
possibility
of
N
-soft
set
as
an
e
xtended
model
of
soft
set,
in
request
to
depict
the
importance
of
e
v
aluations,
in
actuality
.
The
concept
of
fuzzy
sets
to
algebra
and
de
v
eloped
a
group
of
fuzzy
subgroup
in
[10].
In
[11],
introduced
the
concept
of
le
v
el
subsets,
which
[12]
used
earlier
.
T
o
sho
w
that
its
le
v
el
subgroups
can
define
almost
all
of
the
global
notions
of
the
fuzzy
subgroup.
The
normal
fuzzy
soft
group
concept,
also
described
a
normal
fuzzy
soft
subgroup’
s
le
v
el
subsets,
and
discussed
some
o
f
its
properties
[13].
The
normal
fuzzy
soft
group
definition
and
described
a
normal
fuzzy
soft
subgroup’
s
le
v
el
subsets
[14].
In
[15],
define
the
intuitionist
fuzzy
group.
In
[16],
attempts
to
research
some
algebraic
e
xistence
of
intuitionistic
fuzzy
subgroups
and
their
properties
with
the
assistance
of
t
h
e
ir
(
;
)
-cut
sets.
Soft
group
and
their
basic
properties
in
[17].
Abelian
soft
group,
soft
group
coset,
normal
soft
group,soft
group
product,
c
yclic
soft
group.
After
that,
soft
f
actor
group,
normal
maximum
soft
group,
simple
soft
group
are
defined
and
some
important
results
are
pro
v
ed
[18].
In
[19],
e
xtended
the
intuiti
v
e
Fuzzy
Soft
Set
to
semigroup.
Some
remarkable
w
ork
on
the
e
xtension
of
fuzzy
sets
and
fuzzy
algebraic
structure
ha
v
e
been
in
v
estig
ated
[20]
-
[28].
The
intention
of
paper
is
to
inaugurate
Pythagorean
fuzzy
N
-soft
groups
as
the
generalization
of
intuitionistic
fuzzy
group
and
soft
group.
So
as
to
tak
e
care
of
this
present
reality
issues
that
intuitionistic
fuzzy
soft
group
unable
to
manage
the
circumstance
that
the
total
of
enrollment
de
gree
and
non-participation
l
e
v
el
of
the
parameter
is
bigger
than
1.
It
mak
es
the
restricted
,
and
af
fects
the
ideal
choice.
Pythagorean
fuzzy
N-soft
set
gi
v
es
an
enormous
number
of
uses
to
the
for
such
genuine
w
orld
issues.
The
paper
is
sorted
out
as:
Section
2
introduces
our
ne
wly
model
of
Pythagorean
fuzzy
N-soft
sets
to
deal
with
the
situations
in
which
we
need
to
gi
v
e
Pythagorean
fuzzy
characterization
and
non-binary
e
v
aluations
of
objects
of
uni
v
erse
with
respect
to
the
parameters.
Lik
e
wise
established
basic
tasks
and
crucial
properties
of
of
Pythagorean
fuzzy
N-soft
sets.
Section
3
gi
v
es
three
dif
ferent
kinds
of
complements
of
Pythagorean
fuzzy
N-soft
sets
and
e
xamined
a
fe
w
outcomes
not
hold
in
Pythagorean
fuzzy
N-soft
sets
complements
as
the
y
hold
in
crisp
set
h
ypothesis
with
the
assistance
of
counter
e
xamples.
Section
4
we
further
talk
ed
about
(
;
;
)
-
cut
of
Pythagorean
fuzzy
N-soft
set
and
their
properties.
Section
5
we
introduced
the
notions
of
Pythagorean
fuzzy
N-soft
groups.
W
e
lik
e
wise
talk
about
some
essential
properties
of
Pythagorean
fuzzy
N-soft
groups
lik
e
groupoid,
normal
group,
left
and
right
cosets,
(
;
;
)
-cut
subgroups
and
some
fundamental
outcomes
identified
with
these
terms.
Finally
,
At
last,
a
solid
end
is
summed
up
in
section
6.
2.
PYTHA
GOREAN
FUZZY
N-SOFT
SET
In
this
portion,
we
concentrate
some
essential
ideas,
principal
properties
and
arithmetic
operations
on
Pytha
gor
ean
fuzzy
N-soft
set
(PFNS
set).
Definition
2..1
Let
X
be
a
univer
sal
set
and
E
be
the
collection
of
par
ameter
s.
Let
A
E
be
non-empty
and
P
F
X
denote
the
collection
of
all
Pytha
gor
ean
fuzzy
subsets
o
ver
X
.
Let
L
=
f
0
;
1
;
2
;
;
N
1
g
be
the
set
of
gr
ading
values.
A
(PFNS
set)
on
X
is
e
xpr
essed
as
(
;
A;
N
)
or
N
A
,
wher
e
:
A
!
P
(
P
F
X
L
)
is
a
mapping
i.e
.
(
;
A;
N
)
=
n
e;
;
l
e
(
)
;
(
e
(
)
;
e
(
))
:
e
2
A;
2
X
;
l
e
(
)
2
L
o
wher
e
e
:
X
!
[0
;
1]
and
e
:
X
!
[0
;
1]
ar
e
mappings
as
well
as
the
pr
operty
0
2
e
(
)
+
2
e
(
)
1
Pytha
gor
ean
fuzzy
N-Soft
gr
oups
(M.
Shazib
Hameed)
Evaluation Warning : The document was created with Spire.PDF for Python.
1032
r
ISSN:
2502-4752
In
particular
,
e
(
)
r
epr
esents
the
de
gr
ee
of
member
ship,
e
(
)
denotes
de
gr
ee
of
non-member
ship
and
l
e
(
)
denotes
the
gr
ading
value
of
the
element
2
X
corr
esponding
to
the
attrib
ute
e
2
A
to
the
set
(
;
A;
N
)
.
The
set
of
all
PFNSSs
o
ver
X
and
with
set
of
par
ameter
s
fr
om
E
is
called
PFNS
class
and
is
designated
as
P
F
N
S
(
X
;
E
;
N
)
.
If
we
write
a
ij
=
e
j
(
i
)
,
b
ij
=
e
j
(
i
)
and
c
ij
=
l
e
j
(
i
)
wher
e
i
=
i;
2
:::;
m
and
i
=
i;
2
:::;
m
,
then
the
PFNS
set
N
A
may
be
e
xpr
essed
in
tab
ular
form
as
N
A
e
1
e
2
e
n
1
c
11
;
(
a
11
;
b
11
)
c
12
;
(
a
12
;
b
12
)
c
1
n
;
(
a
1
n
;
b
1
n
)
2
c
21
;
(
a
21
;
b
21
)
c
22
;
(
a
22
;
b
22
)
c
2
n
;
(
a
2
n
;
b
2
n
)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
m
c
m
1
;
(
a
m
1
;
b
m
1
)
c
m
2
;
(
a
m
2
;
b
m
2
)
c
mn
;
(
a
mn
;
b
mn
)
and
in
matrix
form
as
(
;
A;
N
)
=
c
ij
;
(
a
ij
;
b
ij
)
m
n
=
0
B
B
B
@
c
11
;
(
a
11
;
b
11
)
c
12
;
(
a
12
;
b
12
)
c
1
n
;
(
a
1
n
;
b
1
n
)
c
21
;
(
a
21
;
b
21
)
c
22
;
(
a
22
;
b
22
)
c
2
n
;
(
a
2
n
;
b
2
n
)
.
.
.
.
.
.
.
.
.
.
.
.
c
m
1
;
(
a
m
1
;
b
m
1
)
c
m
2
;
(
a
m
2
;
b
m
2
)
c
mn
;
(
a
mn
;
b
mn
)
1
C
C
C
A
This
matrix
is
r
eferr
ed
to
as
the
Pytha
gor
ean
fuzzy
N-soft(PFNS)
matrix.
Definition
2..2
A
PFNS
set
N
A
o
ver
X
is
known
as
null
PFNS
set,
symbolized
as
0
and
defined
as
0
=
n
e;
;
l
(
)
;
(
(
)
;
(
))
:
e
2
A;
2
X
;
l
(
)
2
L
o
wher
e
(
)
=
0
;
(
)
=
1
and
l
(
)
=
0
:
Definition
2..3
A
PFNS
set
N
A
o
ver
X
is
said
to
be
absolute
PFNS
set,
symbolized
as
N
1
E
and
defined
as
N
1
E
=
n
e;
;
l
E
(
)
;
(
E
(
)
;
E
(
))
:
e
2
E
;
2
X
;
l
E
(
)
2
L
o
wher
e
E
(
)
=
1
;
E
(
)
=
0
and
l
E
(
)
=
N
1
:
Example
2..4
Let
X
=
f
i
:
i
=
1
;
2
;
;
7
g
,
E
=
f
e
i
:
i
=
1
;
2
;
;
5
g
and
L
=
f
0
;
1
;
2
;
;
8
g
.
Assume
that
A
=
f
e
1
;
e
3
g
.
Then,
9
A
=
n
e
1
;
1
;
4
;
(0
:
3
;
0
:
9)
;
4
;
6
;
(0
:
5
;
0
:
2)
;
7
;
8
;
(0
:
7
;
0
:
4)
;
e
3
;
2
;
3
;
(0
:
2
;
0
:
8)
;
3
;
2
;
(0
:
1
;
0
:
6)
;
5
;
2
;
(0
:
3
;
0
:
7)
o
is
a
PF9S
set
o
ver
X
.
The
r
epr
esentation
in
tab
ular
form
of
9
A
is
9
A
e
1
e
2
e
3
e
4
e
5
1
4,(0.3,
0.9)
0,(0,
1)
0,(0,
1)
0,(0,
1)
0,(0,
1)
2
0,(0,
1)
0,(0,
1)
3,(0.2,
0.8)
0,(0,
1)
0,(0,
1)
3
0,(0,
1)
0,(0,
1)
2,(
0.1,
0.6)
0,(0,
1)
0,(0,
1)
4
6,(0.5,
0.2)
0,(0,
1)
0,(0,
1)
0,(0,
1)
0,(0,
1)
5
0,(0,
1)
0,(0,
1)
2,
(0.3,
0.7)
0,(0,
1)
0,(0,
1)
6
0,(0,
1)
0,(0,
1)
0,(0,
1)
0,(0,
1)
0,(0,
1)
7
8,(
0.7,
0.4)
0,(0,
1)
0,(0,
1)
0,(0,
1)
0,(0,
1)
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
21,
No.
2,
February
2021
:
1030
–
1038
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1033
The
corr
esponding
PF9S
matrix
is
(
;
A;
9)
=
[
a
ij
;
b
ij
]
7
5
=
0
B
B
B
B
B
B
B
B
@
4
;
(0
:
3
;
0
:
9)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
3
;
(0
:
2
;
0
:
8)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
2
;
(0
:
1
;
0
:
6)
0
;
(0
;
1)
0
;
(0
;
1)
6
;
(0
:
5
;
0
:
2)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
2
;
(0
:
3
;
0
:
7)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
8
;
(0
:
7
;
0
:
4)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
0
;
(0
;
1)
1
C
C
C
C
C
C
C
C
A
Definition
2..5
Let
N
1
A
and
N
2
B
be
PFNS
set
o
ver
X
.
Then
their
e
xtended
union
is
defined
as
N
C
=
N
1
A
f
[
E
N
2
B
;
N
C
=
(
;
C
;
N
)
=
n
e;
;
l
C
(
)
;
(
C
(
)
;
C
(
))
:
e
2
C
;
2
X
;
l
C
(
)
2
L
o
wher
e
,
C
=
A
[
B
;
N
=
max
f
N
1
;
N
2
g
;
C
(
)
=
max
f
A
(
)
;
B
(
)
g
;
C
(
)
=
min
f
A
(
)
;
B
(
)
g
and
l
C
(
)
=
m
ax
f
l
A
(
)
;
l
B
(
)
g
8
e
2
C
:
Definition
2..6
Let
N
1
A
and
N
2
B
be
PFNS
set
o
ver
X
.
Then
their
r
estricted
union
is
defined
as
N
C
=
N
1
A
f
[
R
N
2
B
;
N
D
=
(
;
D
;
N
)
=
n
e;
;
l
D
(
)
;
(
D
(
)
;
D
(
))
:
e
2
D
;
2
X
;
l
D
(
)
2
L
o
wher
e
,
D
=
A
\
B
;
N
=
max
f
N
1
;
N
2
g
;
D
(
)
=
max
f
A
(
)
;
B
(
)
g
;
D
(
)
=
min
f
A
(
)
;
B
(
)
g
and
l
D
(
)
=
max
f
l
A
(
)
;
l
B
(
)
g
8
e
2
D
:
Definition
2..7
Let
N
1
A
and
N
2
B
be
PFNS
set
o
ver
X
.
Then
their
e
xtended
inter
section
is
defined
as
N
C
=
N
1
A
f
\
E
N
2
B
;
N
C
=
(
;
C
;
N
)
=
n
e;
;
l
C
(
)
;
(
C
(
)
;
C
(
))
:
e
2
C
;
2
X
;
l
C
(
)
2
L
o
wher
e
,
C
=
A
[
B
;
N
=
min
f
N
1
;
N
2
g
;
C
(
)
=
min
f
A
(
)
;
B
(
)
g
;
C
(
)
=
max
f
A
(
)
;
B
(
)
g
and
l
C
(
)
=
m
in
f
l
A
(
)
;
l
B
(
)
g
8
e
2
C
:
Definition
2..8
Let
N
1
A
and
N
2
B
be
PFNS
set
o
ver
X
.
Then
their
r
estricted
inter
section
is
defined
as
N
C
=
N
1
A
f
\
R
N
2
B
;
N
D
=
(
;
D
;
N
)
=
n
e;
;
l
D
(
)
;
(
D
(
)
;
D
(
))
:
e
2
D
;
2
X
;
l
D
(
)
2
L
o
wher
e
,
D
=
A
\
B
;
N
=
min
f
N
1
;
N
2
g
;
D
(
)
=
min
f
A
(
)
;
B
(
)
g
;
D
(
)
=
max
f
A
(
)
;
B
(
)
g
and
l
D
(
)
=
m
i
n
f
l
A
(
)
;
l
B
(
)
g
8
e
2
D
:
Proposition
2..9
If
N
1
A
and
N
2
B
ar
e
PFNS
sets
o
ver
X
,
then
(i)
N
1
A
f
\
E
N
2
B
e
N
1
A
e
N
1
A
f
[
E
N
2
B
(ii)
N
1
A
f
\
E
N
2
B
e
N
2
B
e
N
1
A
f
[
E
N
2
B
.
(iii)
N
1
A
f
\
R
N
2
B
e
N
1
A
e
N
1
A
f
[
E
N
2
B
(i
v)
N
1
A
f
\
R
N
2
B
e
N
2
B
e
N
1
A
f
[
E
N
2
B
.
Pytha
gor
ean
fuzzy
N-Soft
gr
oups
(M.
Shazib
Hameed)
Evaluation Warning : The document was created with Spire.PDF for Python.
1034
r
ISSN:
2502-4752
3.
COMPLEMENTS
OF
PYTHA
GOREAN
N-SOFT
SET
Definition
3..1
Let
N
A
be
PFNS
set
o
ver
X
.
Then
their
weak
complement
is
r
epr
esented
as
N
A
w
c
and
defined
by
N
A
w
c
=
n
e;
;
l
w
c
A
(
)
;
(
w
c
A
(
)
;
w
c
A
(
))
:
e
2
A;
2
X
;
l
w
c
A
(
)
2
L
o
wher
e
,
w
c
A
(
)
=
A
(
)
;
w
c
A
(
)
=
A
(
)
and
l
w
c
A
(
)
\
l
A
(
)
=
8
e
2
A:
Definition
3..2
Let
N
A
be
PFNS
set
o
ver
X
.
Then
their
top
weak
complement
is
r
epr
esented
as
N
A
tw
c
and
defined
by
N
A
tw
c
=
n
e;
;
l
tw
c
A
(
)
;
(
tw
c
A
(
)
;
tw
c
A
(
))
:
e
2
A;
2
X
;
l
tw
c
A
(
)
2
L
o
wher
e
,
tw
c
A
(
)
=
A
(
)
;
tw
c
A
(
)
=
A
(
)
and
l
tw
c
A
(
)
=
(
N
1
;
if
l
A
(
)
<
N
1
0
;
if
l
A
(
)
=
N
1
8
e
2
A:
Definition
3..3
Let
N
A
be
PFNS
set
o
ver
X
.
Then
their
bottom
weak
complement
is
r
epr
esented
as
N
A
bw
c
and
defined
by
N
A
bw
c
=
n
e;
;
l
bw
c
A
(
)
;
(
bw
c
A
(
)
;
bw
c
A
(
))
:
e
2
A;
2
X
;
l
bw
c
A
(
)
2
L
o
wher
e
,
bw
c
A
(
)
=
A
(
)
;
bw
c
A
(
)
=
A
(
)
and
l
bw
c
A
(
)
=
(
N
1
;
if
l
A
(
)
=
0
0
;
if
l
A
(
)
>
0
8
e
2
A:
Theorem
3..4
Let
N
A
be
PFNS
set,
0
be
null
PFNS
and
N
1
E
absolute
PFNS
o
v
er
X
,
then
follo
wing
results
that
hold
in
crisp
set
theory
b
ut
not
hold
in
PFNS
set
theory
(i)
0
w
c
6
=
N
1
E
:
(ii)
N
1
E
w
c
6
=
0
.
(iii)
N
A
w
c
w
c
6
=
N
A
Pr
oposition
3..5
Let
N
A
be
PFNS
set,
0
be
null
PFNS
and
N
1
E
absolute
PFNS
o
v
er
X
,
then
(i)
0
tw
c
=
N
1
E
:
(ii)
N
1
E
tw
c
=
0
.
Theor
em
3..6
Let
N
A
be
PFNS
set,
o
v
er
X
,
then
(i)
N
A
bw
c
bw
c
6
=
N
A
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
21,
No.
2,
February
2021
:
1030
–
1038
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1035
4.
(
;
;
)
-CUT
OF
PYTHA
GOREAN
FUZZY
N-SOFT
SET
AND
THEIR
PR
OPER
TIES
Definition
4..1
Let
N
A
be
a
PFNS
set
on
X
.
Then
(
;
;
)
-cut
of
N
A
is
a
crisp
subset
C
;
;
(
N
A
)
of
PFNS
N
A
is
given
by
C
;
;
(
N
A
)
=
i
:
i
2
X
suc
h
that
A
(
i
)
;
A
(
i
)
;
l
A
(
i
)
wher
e
;
2
[0
;
1]
with
0
2
+
2
1
and
2
f
0
;
1
;
2
;
N
1
g
.
Pr
oposition
4..2
If
N
A
and
N
B
be
two
PFNS
set’
s
on
X
,
then
following
holds.
1.
C
;
;
(
N
A
)
C
;
;
(
N
A
)
if
;
and
2.
C
1
;
;
(
N
A
)
C
;
;
(
N
A
)
C
;
1
;
(
N
A
)
3.
N
A
N
B
)
C
;
;
(
N
A
)
C
;
;
(
N
B
)
4.
C
;
;
(
N
A
f
[
E
N
B
)
C
;
;
(
N
A
)
f
[
E
C
;
;
(
N
B
)
5.
C
;
;
(
N
A
f
\
R
N
B
)
=
C
;
;
(
N
A
)
f
\
R
C
;
;
(
N
B
)
equality
hold
if
2
+
2
=
1
6.
C
;
;
(
f
\
R
N
A
i
)
=
f
\
R
C
;
;
(
N
A
i
)
7.
C
0
;
1
;
0
(
N
A
)
=
X
5.
PYTHA
GOREAN
FUZZY
N-SOFT
GR
OUPS
In
this
section,
we
introduce
notion
of
p
ythagorean
fuzzy
N-soft
group
(PFNS-group)
by
i
nspiring
from
the
Intuitionistic
fuzzy
soft
groups
of
Karaas
lan
et.al
[23].
In
this
section
G
denotes
an
arbitrary
group
with
identity
element
e
.
Definition
5..1
Let
G
be
a
gr
oup
and
N
A
be
a
PFNS
set
on
X
.
(
N
A
)
G
is
called
PFNS-gr
oupoid
if
(
N
A
)
G
(
i
:
j
)
e
(
N
A
)
G
(
i
)
u
(
N
A
)
G
(
j
)
8
i
;
j
2
G
.
i.e
.
A
(
i
:
j
)
A
(
i
)
^
A
(
j
)
;
A
(
i
:
j
)
A
(
i
)
_
A
(
j
)
and
l
A
(
i
:
j
)
l
A
(
i
)
^
l
A
(
j
)
8
i
;
j
2
G
:
Definition
5..2
Let
G
be
a
gr
oup
and
N
A
be
a
PFNS
set
on
X
.
(
N
A
)
G
is
called
PFNS-gr
oup
if
(
N
A
)
G
(
i
:
j
)
e
(
N
A
)
G
(
i
)
u
(
N
A
)
G
(
j
)
and
(
N
A
)
G
(
1
i
)
e
(
N
A
)
G
(
i
)
8
i
;
j
2
G
.
i.e
.
A
(
i
:
j
)
A
(
i
)
^
A
(
j
)
;
A
(
1
i
)
A
(
i
)
;
A
(
i
:
j
)
A
(
i
)
_
A
(
j
)
;
A
(
1
i
)
A
(
i
)
and
l
A
(
i
:
j
)
l
A
(
i
)
^
l
A
(
j
)
8
i
;
j
2
G
:
l
A
(
1
i
)
l
A
(
i
)
:
Example
5..3
Let
S
3
=
<
;
j
3
=
2
=
(
)
2
=
e
>
be
a
symmetric
gr
oup
w
.r
.t
multiplication,
X
=
A
=
f
x
1
;
x
2
;
x
3
;
x
4
;
x
5
;
x
6
g
and
7
A
be
a
PF7S
set
on
X
given
below
.
(
7
A
)
S
3
e
2
2
x
1
3
;
(0
:
2
;
0
:
2)
3
;
(0
:
2
;
0
:
4)
3
;
(0
:
2
;
0
:
4)
3
;
(0
:
2
;
0
:
3)
3
;
(0
:
2
;
0
:
4)
3
;
(0
:
2
;
0
:
4)
x
2
5
;
(0
:
6
;
0
:
3)
5
;
(0
:
6
;
0
:
5)
5
;
(0
:
6
;
0
:
5)
5
;
(0
:
6
;
0
:
4)
5
;
(0
:
6
;
0
:
5)
5
;
(0
:
6
;
0
:
5)
x
3
6
;
(0
:
9
;
0
:
1)
6
;
(0
:
9
;
0
:
3)
6
;
(0
:
9
;
0
:
3)
6
;
(0
:
9
;
0
:
2)
6
;
(0
:
9
;
0
:
3)
6
;
(0
:
9
;
0
:
3)
x
4
4
;
(0
:
5
;
0
:
4)
4
;
(0
:
5
;
0
:
6)
4
;
(0
:
5
;
0
:
6)
4
;
(0
:
5
;
0
:
5)
4
;
(0
:
5
;
0
:
6)
4
;
(0
:
5
;
0
:
6)
x
5
2
;
(0
:
2
;
0
:
5)
2
;
(0
:
2
;
0
:
5)
2
;
(0
:
2
;
0
:
5)
2
;
(0
:
2
;
0
:
6)
2
;
(0
:
2
;
0
:
5)
2
;
(0
:
2
;
0
:
5)
x
6
1
;
(0
:
1
;
0
:
7)
1
;
(0
:
1
;
0
:
9)
1
;
(0
:
1
;
0
:
9)
1
;
(0
:
1
;
0
:
8)
1
;
(0
:
1
;
0
:
9)
1
;
(0
:
1
;
0
:
9)
whic
h
satisfied
all
the
condition
of
(
7
A
)
S
3
Definition
5..4
Let
G
be
a
gr
oup
and
N
A
be
a
PFNS
set
on
X
.
(
N
A
)
G
is
called
PFNS
normal
gr
oup
if
A
(
i
:
j
)
=
A
(
j
:
i
)
;
A
(
i
:
j
)
=
A
(
j
:
i
)
;
l
A
(
i
:
j
)
=
l
A
(
j
:
i
)
8
i
;
j
2
G
:
(
7
A
)
S
3
is
not
PF7S
normal
group
because
x
1
(
:
2
)
=
0
:
4
,
x
1
(
2
:
)
=
0
:
3
:
This
implies
x
1
(
:
2
)
6
=
x
1
(
2
:
)
:
Pytha
gor
ean
fuzzy
N-Soft
gr
oups
(M.
Shazib
Hameed)
Evaluation Warning : The document was created with Spire.PDF for Python.
1036
r
ISSN:
2502-4752
Pr
oposition
5..5
A
necessary
and
suf
ficient
condition
for
PFNS
set
N
A
of
a
gr
oup
G
to
be
a
PFNSG
of
G
is
that
8
i
;
j
2
G
;
A
(
i
:
1
j
)
A
(
i
)
^
A
(
j
)
;
A
(
i
:
1
j
)
A
(
i
)
_
A
(
j
)
and
l
A
(
i
:
1
j
)
l
A
(
i
)
^
l
A
(
j
)
:
Pr
oof:
Proof
of
this
preposition
is
ob
vious
from
Definition
5..2.
Example
5..6
Let
S
3
=
<
;
j
3
=
2
=
(
)
2
=
e
>
be
a
symmetric
gr
oup
w
.r
.t
multiplication,
X
=
A
=
f
x
1
;
x
2
;
x
3
;
x
4
;
x
5
;
x
6
g
and
7
A
be
a
PF7S
subset
H
on
X
given
below
.
(
7
A
)
H
e
2
x
1
3
;
(0
:
2
;
0
:
2)
3
;
(0
:
2
;
0
:
4)
3
;
(0
:
2
;
0
:
4)
x
2
5
;
(0
:
6
;
0
:
3)
5
;
(0
:
6
;
0
:
5)
5
;
(0
:
6
;
0
:
5)
x
3
6
;
(0
:
9
;
0
:
1)
6
;
(0
:
9
;
0
:
3)
6
;
(0
:
9
;
0
:
3)
x
4
4
;
(0
:
5
;
0
:
4)
4
;
(0
:
5
;
0
:
6)
4
;
(0
:
5
;
0
:
6)
x
5
2
;
(0
:
2
;
0
:
5)
2
;
(0
:
2
;
0
:
5)
2
;
(0
:
2
;
0
:
5)
x
6
1
;
(0
:
1
;
0
:
7)
1
;
(0
:
1
;
0
:
9)
1
;
(0
:
1
;
0
:
9)
whic
h
satisfied
all
the
condition
of
PFNS
normal
subgr
oup
(
7
A
)
H
:
Definition
5..7
Let
(
N
A
)
G
be
PFNSG
of
a
gr
oup
G
.
F
or
any
j
2
G
;
the
PFNS
set
j
:
(
N
A
)
G
is
called
left
coset
of
(
N
A
)
G
if
j
:
(
N
A
)
G
(
i
)
=
(
N
A
)
G
(
j
1
:
i
)
8
i
;
j
2
G
;
i.e
j
:
A
(
i
)
=
A
(
j
1
:
i
)
;
j
:
A
(
i
)
=
A
(
j
1
:
i
)
and
j
:
l
A
(
i
)
=
l
A
(
j
1
:
i
)
:
Definition
5..8
Let
(
N
A
)
G
be
PFNSG
of
a
gr
oup
G
.
F
or
any
j
2
G
;
the
PFNS
set
(
N
A
)
G
:
j
is
called
right
coset
of
(
N
A
)
G
if
(
N
A
)
G
:
j
(
i
)
=
(
N
A
)
G
(
i
:
j
1
)
8
i
;
j
2
G
;
i.e
A
(
i
)
:
j
=
A
(
i
:
j
1
)
;
A
(
i
)
:
j
=
A
(
i
:
j
1
)
and
l
A
(
i
)
:
j
=
l
A
(
i
:
j
1
)
:
Pr
oposition
5..9
A
necessary
and
suf
ficient
condition
for
PFNSG
(
N
A
)
G
of
a
gr
oup
G
to
be
a
PFNS
normal
gr
oup
of
G
is
that
8
i
;
j
2
G
;
A
(
j
1
:
i
:
j
)
=
A
(
i
)
;
A
(
j
1
:
i
:
j
)
=
A
(
i
)
and
l
A
(
j
1
:
i
:
j
)
=
l
A
(
i
)
:
Pr
oof:
Let
j
2
G
be
an
y
element,
then
A
(
j
1
:
i
:
j
)
=
A
(
j
1
:
(
i
:
j
))
=
A
(
i
:
(
j
1
:
j
))
=
A
(
i
:
e
)
=
A
(
i
)
;
A
(
j
1
:
i
:
j
)
=
A
(
j
1
:
(
i
:
j
))
=
A
(
i
:
(
j
1
:
j
))
=
A
(
i
:
e
)
=
A
(
i
)
and
l
A
(
j
1
:
i
:
j
)
=
l
A
(
j
1
:
(
i
:
j
))
=
l
A
(
i
:
(
j
1
:
j
))
=
l
A
(
i
:
e
)
=
l
A
(
i
)
:
If
(
N
A
)
G
be
PFNS
normal
subgroup
of
a
group
G
.
Then
it
is
ob
vious
that
PFNS
left
cos
et
of
(
N
A
)
G
and
PFNS
right
coset
of
(
N
A
)
G
are
equal.
In
this
situation
we
can
call
PFNS
coset
instead
of
PFNS
left
coset
or
PFNS
right
coset.
Pr
oposition
5..10
Let
G
be
a
gr
oup.
Then
(
N
A
)
G
=
n
e;
;
l
e
(
)
;
(
e
(
)
;
e
(
))
;
2
G
:
l
e
(
)
=
l
e
(
e
)
;
e
(
)
=
e
(
e
)
;
e
(
)
=
e
(
e
)
o
is
PFNS
normal
subgr
oup
of
G
:
Pr
oof:
It
is
ob
vious.
Pr
oposition
5..11
Let
(
N
A
)
G
be
PFNSG
of
a
gr
oup
G
.
Then
C
;
;
(
N
A
)
is
a
subgr
oup
of
gr
oup
G
,
wher
e
A
(
e
)
;
A
(
e
)
;
l
A
(
e
)
and
e
is
the
identity
of
G
:
Pr
oof:
Let
i
;
j
2
C
;
;
(
N
A
)
be
any
two
elements.
Then
A
(
i
)
;
A
(
i
)
and
l
A
(
i
)
and
A
(
j
)
;
A
(
j
)
and
l
A
(
j
)
;
)
A
(
i
)
^
A
(
j
)
;
A
(
i
)
_
A
(
j
)
and
l
A
(
i
)
^
l
A
(
j
)
:
As
(
N
A
)
G
be
PFNSG
of
a
gr
oup
G
.
Ther
e-
for
e
,
A
(
i
:
1
j
)
A
(
i
)
^
A
(
j
)
;
A
(
i
:
1
j
)
A
(
i
)
_
A
(
j
)
and
l
A
(
i
:
1
j
)
l
A
(
i
)
^
l
A
(
j
)
:
Thus
i
:
1
j
2
C
;
;
(
N
A
)
:
Hence
C
;
;
(
N
A
)
is
a
subgr
oup
of
G
:
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
21,
No.
2,
February
2021
:
1030
–
1038
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
1037
Pr
oposition
5..12
Let
(
N
A
)
G
be
PFNS
normal
subgr
oup
of
a
gr
oup
G
.
Then
C
;
;
(
N
A
)
is
a
normal
sub-
gr
oup
of
gr
oup
G
,
wher
e
A
(
e
)
;
A
(
e
)
;
l
A
(
e
)
and
e
is
the
identity
of
G
:
Pr
oof:
Let
i
2
C
;
;
(
N
A
)
and
i
2
G
be
any
element.
Then
A
(
i
)
;
A
(
i
)
and
l
A
(
i
)
:
Also
(
N
A
)
G
be
PFNS
normal
subgr
oup
of
a
gr
oup
G
.
Ther
e-
for
e
,
A
(
1
j
:
i
:
j
)
=
A
(
i
;
A
(
1
j
:
i
:
j
)
=
A
(
i
)
and
l
A
(
1
j
:
i
:
j
)
=
l
A
(
i
)
:
This
implies
that
A
(
1
j
:
i
:
j
)
=
A
(
i
)
;
A
(
1
j
:
i
:
j
)
=
A
(
i
)
and
l
A
(
1
j
:
i
:
j
)
=
l
A
(
1
i
)
:
Thus
1
j
:
i
:
j
2
C
;
;
(
N
A
)
:
Hence
C
;
;
(
N
A
)
is
a
normal
subgr
oup
of
G
:
Example
5..13
Let
S
3
=
<
;
j
3
=
2
=
(
)
2
=
e
>
be
a
symmetric
gr
oup
w
.r
.t
multiplication,
X
=
A
=
f
x
1
;
x
2
;
x
3
;
x
4
;
x
5
;
x
6
g
and
7
A
be
a
PF7S
subset
H
on
X
given
below
.
C
0
:
1
;
0
:
7
;
1
(
7
A
)
e
2
x
1
3
;
(0
:
2
;
0
:
2)
3
;
(0
:
2
;
0
:
4)
3
;
(0
:
2
;
0
:
4)
x
2
5
;
(0
:
6
;
0
:
3)
5
;
(0
:
6
;
0
:
5)
5
;
(0
:
6
;
0
:
5)
x
3
6
;
(0
:
9
;
0
:
1)
6
;
(0
:
9
;
0
:
3)
6
;
(0
:
9
;
0
:
3)
x
4
4
;
(0
:
5
;
0
:
4)
4
;
(0
:
5
;
0
:
6)
4
;
(0
:
5
;
0
:
6)
x
5
2
;
(0
:
2
;
0
:
5)
2
;
(0
:
2
;
0
:
5)
2
;
(0
:
2
;
0
:
5)
x
6
1
;
(0
:
1
;
0
:
7)
1
;
(0
:
1
;
0
:
9)
1
;
(0
:
1
;
0
:
9)
whic
h
satisfied
all
the
condition
of
PFNS
normal
subgr
oup
(
7
A
)
H
:
Hence
C
0
:
1
;
0
:
7
;
1
(
7
A
)
is
normal
subgr
oup.
Pr
oposition
5..14
If
(
N
A
)
G
and
(
N
B
)
G
be
two
PFNSG’
s
of
a
gr
oup
G
,
then
(
N
A
)
G
T
(
N
B
)
G
is
also
a
PFNSG
of
a
gr
oup
G
.
6.
CONCLUSION
In
this
article,
we
introduced
no
v
el
concept
of
p
ythagorean
fuzzy
N
soft
sets
(PFNS
sets).
The
purpose
of
this
study
is
to
lay
the
foundation
of
inno
v
ati
v
e
theory
of
PFNS
sets
including
basic
algebraic
oper
-
ations
lik
e
e
xtended
and
restricted
union,
e
xtended
and
restricted
intersection
and
three
types
of
complements
(weak
complement,
top
weak
complement
and
bottom
weak
complement)
on
them
and
some
results
of
pri-
mary
importance
by
using
these
operations.
A
lar
ge
number
of
illustrations
are
included
with
each
definition
to
comprehend
the
notions
ef
fecti
v
ely
.
Furthermore,
we
proposed
(
;
;
)
cut
PFNS
sets,
we
elaborate
some
of
its
basic
properties.
W
e
al
so
initiated
the
notion
of
p
ythagorean
fuzzy
N
soft
groups
(PFNS-groups)
and
presented
certain
properties
of
PFNS-groups.
PFNS-groups
is
the
e
xtension
of
soft
groups,
fuzzy
soft
groups,
intuitionistic
soft
groups,
p
ythagorean
soft
groups,
N-soft
groups.
Later
on,
we
intend
to
e
xtend
o
v
er
research
to
the
ne
w
operational
la
ws
and
properties
on
PFNS-groups
and
plane
to
apply
more
w
ays
to
deal
with
dif
ferent
h
ybrid
models
including
neutrosophic
N-soft
groups,
bipolar
N-soft
groups
and
m-polar
N-soft
groups
etc.
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L.
A.
Zadeh,
“Fuzzy
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Information
and
Contr
ol,
v
ol.
8,
pp.
338-353,
1965.
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K.
T
.
Atanasso
v
and
S.
Stoe
v
a,
“Intuitionistic
fuzzy
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P
olish
Symp.
on
Int
erval
and
Fuzzy
Mathe-
matics
,
Poznan,
pp.
23-26,
1983.
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K.
T
.
Atanasso
v
,
“Intuitionistic
fuzzy
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ol.
20,
pp.
87-96,
1986.
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D.
Molodtso
v
,
“Soft
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Computer
s
and
Mathematics
with
Applications
,
37
(1999),
19-31.
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R.
R.
Y
ager
,
“Pythagorean
membership
grades
in
multi-criteria
decision
making”,
IEEE
T
r
ansactions
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Fuzzy
Systems
,
v
ol.
22,
pp.958-965,
2014.
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X.D.
Peng,
Y
.
Y
ang,
“Some
results
for
Pythagorean
fuzzy
sets”,
International
J
ournal
of
Intellig
ent
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tems
,
v
ol.
30,
pp
1133-1160,
2015.
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X.D.
Peng,
H.Y
.
Y
uan
and
Y
.
Y
ang,
“Pythagorean
fuzzy
information
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and
their
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International
J
ournal
of
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ent
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v
ol.
32,
pp.
991-1029,
2017.
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X.D.
Peng,Y
.
Y
.
Y
ang,
J.
Song
and
Y
.
Jiang,
“Pythagorean
Fuzzy
Soft
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and
Its
Application”,
Computer
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