In this paper, two new Operator defined over IVIFSs were introduced, which will be “Multiplication of an IVIFS
 with  and Multiplication of an IVIFS 
  with the natural number   ”are proved.

1. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
TWO NEW OPERATOR DEFINED OVER
INTERVAL VALUED INTUITIONISTIC FUZZY
SETS
S. Sudharsan1, 2 and D. Ezhilmaran3
1Research Scholar, Bharathiar University, Coimbatore -641046, India.
2Department of Mathematics, C. Abdul Hakeem College of Engineering & Technology,
Melvisharam,Vellore – 632 509, Tamilnadu, India.
3School of Advanced Sciences, VIT University, Vellore – 632 014, Tamilnadu, India.
ABSTRACT
In this paper, two new Operator defined over IVIFSs were introduced, which will be “Multiplication of an
IVIFS with and Multiplication of an IVIFS
with the natural number
”are proved.
Key Words:
Intuitionistic fuzzy set, Interval valued Intuitionistic fuzzy sets, Operations over interval valued
intuitionistic fuzzy sets.
AMS CLASSIFICATION: 03E72
1. INTRODUCTION
In 1965, Fuzzy sets theory was proposed by L. A. Zadeh[18]. In 1986, the concept of
intuitionistic fuzzy sets (IFSs), as a generalization of fuzzy set were introduced by K.
Atanassov[1]. After the introduction of IFS, many researchers have shown interest in the IFS
theory and applied in numerous fields, such as pattern recognition, machine learning, image
processing, decision making and etc... In 1994, new operations defined over the intuitionistic
fuzzy sets was proposed by K. Atanassov[3]. In 2000, Some operations on intuitionistic fuzzy
sets were proposed by Supriya Kumar De, Ranjit Biswas and Akhil Ranjan Roy[12]. In 2001, an
application of intuitionistic fuzzy sets in medical diagnosis were proposed by Supriya Kumar De,
Ranjit Biswas and Akhil Ranjan Roy [13]. In 2006, n-extraction operation over intuitionistic
fuzzy sets were proposed by B. Riecan and K. Atanassov[9]. In 2010, Operation division by n
over intuitionistic fuzzy sets were proposed by B. Riecan and K. Atanassov [10]. In 2010,
Remarks on equalities between intuitionistic fuzzy sets was K. Atanassov[4]. In 2008, properties
of some IFS operators and operations were proposed by Liu Q, Ma C and Zhou X [7]. In 2008,
Four equalities connected with intuitionistic fuzzy sets was proposed by T. Vasilev [14]. In 2011,
Intuitionistic fuzzy sets: Some new results were proposed by R. K. Verma and B. D. Sharma[15].
In 1989, the notion of Interval-Valued Intuitionistic Fuzzy Sets which is a generalization of both
DOI : 10.5121/ijfls.2014.4401 1

2. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
Intuitionistic Fuzzy Sets and Interval-Valued Fuzzy Sets were proposed by K. Atanassov and G.
Gargov [5]. After the introduction of IVIFS, many researchers have shown interest in the IVIFS
theory and applied it to the various field. In 1994, Operators over interval-valued intuitionistic
fuzzy sets was proposed by K. Atanassov[6]. In 2007, methods for aggregating interval-valued
intuitionistic fuzzy information and their application to decision making was proposed by Z.S. Xu
[17]. In 2007, Some geometric aggregation operators based on interval-valued intuitionistic fuzzy
sets and their application to group decision making were proposed by G.W .Wei and X.R.Wang
[16]. In 2012, Some Results on Generalized Interval-Valued Intuitionistic Fuzzy Sets were
proposed by Monoranjan Bhowmik and Madhumangal Pal[8]. In 2013, Interval-Valued
Intuitionistic Hesitant Fuzzy Aggregation Operators and Their Application in Group Decision-
Making were proposed by Zhiming Zhang [19]. In 2014, new Operations over Interval Valued
Intuitionistic Hesitant Fuzzy Set were proposed by Broumi and Florentin Smarandache [11]. This
paper proceeds as follows: In section 2 some basic definitions related to intuitionistic fuzzy sets,
interval valued intuitionistic fuzzy sets and set operations are introduced over the IVIFSs are
presented. In section 3 two new operators and
2
defined over IVIFSs are introduced and
proved. In section 4 and 5, Conclusion and Acknowledgments are given.
2. PRELIMINARIES
Definition 2.1:
Intuitionistic Fuzzy Set [1,2]: An intuitionistic fuzzy set A in the finite universe X is defined as

3. , where

4. and

5. with the
condition ! # ! , for any $ The intervals and
denote the degree of membership function and the degree of non-membership of the element x to
the set A.
%
Definition 2.2:
Interval valued Intuitionistic Fuzzy Set [5]: An Interval valued intuitionistic fuzzy set A in the
finite universe X is defined as A

6. . The intervals and
denote the degree of membership function and the degree of non-membership of the element x to
the set A. For every , and are closed intervals and their Left and Right end
points are denoted by

7. ,
%

8. and
.Let us denote
'

9. ( %
)

10. (

11. %

12. ) *Where
#

13. %
+
% + . Especially if %

14. and
%
then the
given IVIFS A is reduced to an ordinary IFS.
Let us define the empty IVIFS, the totally uncertain IVIFS and the unit IVIFS by:
,-

15. .-

16. /01-

17. .
Definition 2.2. Set operations on IVIFSs [5]: Let A and B be two IVIFSs on the universe X,
where

18. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
3
'2

19. ( %
)

20. (

21. %

22. )3 * 4
'2

23. (5 %
)

24. (5 %

25. 5
)3 *

26. 5
Here, we define some set operations for IVIFSs:
6 4 78

27. (9/! %

28. 5 %

29. 9/!

30. 5
)

31. %

32. 5
%

33. 9:!
!9:!

34. 5
;
= 4 78

35. (9:! %

36. 5 %

37. 9:!

38. 5
)

39. %

40. 5
%

41. 9/!
!9/!

42. 5
;
# 4 ?

43. ! %
# 5 %
@ %
5 %

44. 5
@
5

45. %

46. 5
%

47. 5
A B
C 4 ?

48. ! %
5 %

49. 5

50. % # 5
% @
%5
%

51. # 5
@
5
A B
D EF

52. !
%

53. ! %
G H

54. I EF

55. ! %

56. ! @ %

57. G H

58. @
%

59. @
J EF

60. ! @

61. !
%

62. G H
EF

63. ! @ ! @ %

64. @ ! @

65. !!
%

66. !
G H
EF

67. !! %

68. !

69. ! @ ! @
%

70. @ ! @
G H

71. M!
K ?

72. LM! %
N

73. L @ M! @
%

74. @ M! @
NA B
?

75. L @ M! @ %

76. @ M! @
N

77. LM!
%

78. M!
NA B

79. Where + is natural number.
3. TWO NEW OPERATOR OPO AND
Q
O DEFINED OVER IVIFS ARE
Q
OP
INTRODUCED AND PROVED
Two new Operators, defined over IVIFS are introduced, which will be an analogous as of
Operations “extraction” as well as of operation “Multiplication of an IVIFS
with
and
Multiplication of an IVIFS with ”. It has the form for every IVIFS and for every natural
number +
R S S T
S S U
V W W W W W X
] ] ] ] ] ^

80. @ Z @ M!

81. Y @ Z @ M! %
[

82. %
YZ @ M! @

83. Z @ M! @
[
_ S S `
S S a
bc

84. ( @ ! @ ! %

85. @ ! @ !
)

86. %
(! @ ! @

87. ! @ ! @
)
d e

88. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
4
Theorem 3.1. For any two IVIFSs A and B and for every natural number + :
/$ = 4 = 4

89. f$ 6 4 6 4
g$
h
= 4
i
=
4

90. 0$
h
6 4
i
6
4
$
Proof: /.
= 4
Y
EF

91. !! %

92. !

93. ! @ ! @
%

94. @ ! @
G H
=
EF

95. !!5 %

96. !5

97. ! @ ! @ 5
%

98. @ ! @ 5
G H
[
k l
j
U
R T
V W W X

99. (9:!! %

100. !5 %

101. 9:!!

102. !5
)

103. m
%

104. @ ! @ 5
9/! @ ! @
%

105. @ ! @ 5
9/! @ ! @
n
] ] ^
a
_ `
p q
o
R S S T
S S U
V W W W W W X
@ ( @ 9:!! %

106. Y

107. !5 %
)

108. !5
@ ( @ 9:!!
)
[

109. (9/! @ ! @
Y
%

110. @ ! @ 5
%)

111. @ ! @ 5
(9/! @ ! @ 5
)
[
] ] ] ] ] ^
_ S S `
S S a
R S S T
S S U
V W W W W W X

112. Y
@ (9/! @ ! %

113. @ !5 %
)

114. @ !5
@ (9/! @ !
)
[

115. 9/ (! @ ! @
Y

116. ! @ ! @ 5
%
%
)

117. 9/ (! @ ! @

118. ! @ ! @ 5
)
[
] ] ] ] ] ^
_ S S `
S S a
R S S T
S S U
V W W W W W X

119. Y
@ 9/ (! @ ! %

120. ! @ !5 %
)

121. @ 9/ (! @ !

122. ! @ !5
)
[

123. Y
%
9/ (! @ ! @

124. ! @ ! @ 5
)

125. %
9/ (! @ ! @

126. ! @ ! @ 5
)
] ] ] ] ] ^
[
_ S S `
S S a
R S S T
S S U
V W W W W W X

127. Y
9: ( @ ! @ ! %

128. @ ! @ !5 %
)

129. 9: ( @ ! @ !

130. @ ! @ !5
)
[

131. Y
%
9/ (! @ ! @

132. ! @ ! @ 5
)

133. %
9/ (! @ ! @

134. ! @ ! @ 5
)
[
] ] ] ] ] ^
_ S S `
S S a
k k k k l
j
bc

135. ( @ ! @ ! %

136. @ ! @ !
)

137. %
(! @ ! @

138. ! @ ! @
)
d e
=
bc

139. ( @ ! @ !5 %

140. @ ! @ !5
)

141. %
(! @ ! @ 5

142. ! @ ! @ 5
)
p p p p q
d e
o

143. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
5
= 4
Hence /is proved and similarly (b) is proved by analogy.
Proof:g.
h
= 4
i
k k k l
j
N

144. LM
78

145. mLM %
Nn

146. L @ M! @
%

147. @ M! @
N
;
=
N

148. LM5
78

149. mLM5 %
Nn

150. L @ M! @ 5
%

151. @ M! @ 5
N;
p p p q
o
k k k k l
j
R S S T
S S U
V W W W W W W X

152. M5 %

153. m9: LM %
] ] ] ] ] ] ^
N

154. 9: LM

155. M5
Nn

156. k l
j
%

157. @ M! @ 5
9/ L @ M! @
% N

158. @ M! @ 5
9/ L @ M! @
N
p q
o
_ S S `
S S a
p p p p q
o
R S S S S S T
S S S S S U
V W W W W W W W W W W W X

159. k k l
j

160. M5 %
@ Zm @ 9: LM %
Nn

161. M5
@ Zm @ 9: LM
Nn
p p q
o

162. k k l
j
%

163. @ M! @ 5
Z9/ L @ M! @
% N

164. @ M! @ 5
Z9/ L @ M! @
N
p p q ] ] ] ] ] ] ] ] ] ] ] ^
o
S S S S S a
_ S S S S S `
R S S S S S T
S S S S S U
V W W W W W W W W W W W W X

165. k k l
j

166. @ M5 % N
@ Z9/ L @ M %

167. @ M5
@ Z9/ L @ M
N
p p q
o

168. k k k l
j
%
9/ YZ @ M! @
%
[

170. Z @ M! @ 5
9/ YZ @ M! @
[

171. Z @ M! @ 5
p p p q
o
] ] ] ] ] ] ] ] ] ] ] ] ^
_ S S S S S `
S S S S S a

172. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
6
R S S S S S T
S S S S S U
V W W W W W W W W W W W W X

173. k k k l
j
N
@ 9/ YZL @ M %
[

175. Z @ M5 %
N
@ 9/YZL @ M

176. Z @ M5
p p p q
[
o

177. k k k l
j
%
9/ YZ @ M! @
%

178. Z @ M! @ 5
[

179. 9/ YZ @ M! @
[

180. Z @ M! @ 5
p p p q
o
] ] ] ] ] ] ] ] ] ] ] ] ^
_ S S S S S `
S S S S S a
R S S S S S T
S S S S S U
V W W W W W W W W W W W W X

181. k k k l
j
N
9:Y @ ZL @ M %
[

183. @ Z @ M5 %
N
9:Y @ ZL @ M
[

184. @ Z @ M5
p p p q
o

185. k k k l
j
%
9/ YZ @ M! @
%
[

187. Z @ M! @ 5
9/ YZ @ M! @
[

188. Z @ M! @ 5
p p p q
o
] ] ] ] ] ] ] ] ] ] ] ] ^
_ S S S S S `
S S S S S a
k k k k k k k k k l
j
R S T
S U
V W W W W X
)

189. @ M( @ r

190. L @ M( @ r %
) N

191. %
YZ @ M! @

192. Z @ M! @
[
] ] ] ] ^
_ S `
S a
=
R S T
S U
V W W W W X
)

193. @ M( @ r5

194. L @ M( @ r5 %
) N

195. %
YZ @ M! @ 5

196. Z @ M! @ 5
[
] ] ] ] ^
_ S `
S a
p p p p p p p p p q
o
=
4
Hence gis proved and similarly (d) is proved by analogy.

197. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
7
Theorem 3.2. For every IVIFS A and for every natural number + :
/$I
I
sf $J
J
s g$I Is 0$J J .
Proof: (a).
I
I
N

198. LM
78

199. mLM %
Nn

200. L @ M! @
%

201. @ M! @
N;
I
U
c

202. Y @ ZL @ M %
R T
N
N

203. @ ZL @ M
%
[

204. YZ @ M! @

205. Z @ M! @
[d
a
_ `
R S T
S U
V W W W X
)

206. @ M( @ r

207. L @ M( @ r %
) N

208. ) N

209. @ L @ M( @ r
m @ L @ M( @ r %
) Nn
] ] ] ^
_ S `
S a
)

210. @ M( @ r
?

211. L @ M( @ r %
) N

212. LM( @ r %
)

213. M( @ r
) NA B
(3.1)
I
N

214. LM
Ibc8

215. mLM %
Nn

216. L @ M! @
%

217. @ M! @
N
;d e

218. M
?

219. LM %
N

220. L @ M %

221. @ M
NA B
)

222. @ M( @ r
?

223. L @ M( @ r %
) N

224. LM( @ r %
)

225. M( @ r
) N
A B
(3.2)
From (3.1) and (3.2), we get
I
I
Hence (a) is proved and similarly (b) is proved by analogy.
Proof:g.
I
IEF

226. !! %

227. !
%

228. @ ! @

229. ! @ ! @
G H
Itu

230. ( @ ! @ ! %

231. @ ! @ !
)

232. (! @ ! @

233. ! @ ! @
%
)
v w

234. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
8
bc

235. ( @ ! @ ! %

236. @ ! @ !
)

237. h @ ( @ ! @ ! %
)

238. @ ( @ ! @ !
)i
d e
'2

239. ( @ ! @ ! %

240. @ ! @ !
)

241. (! @ ! %

242. ! @ !
)3 *
(3.3)
I
IEF

243. !! %

244. !

245. ! @ ! @
%

246. @ ! @
G H
EF

247. !! %

248. !

249. ! @ ! %

250. @ !
G H
'2

251. ( @ ! @ ! %

252. @ ! @ !
)

253. (! @ ! %

254. ! @ !
)3 *
(3.4)
From (3.3) and (3.4), we get
I I
Hence g is proved and similarly (d) is proved by analogy.
Theorem 3.3. For any two IVIFSs A and B and for every natural number + :
/$
h
# 4
i
#
4

255. f$
h
C4
i
C
4

256. g$ # 4 # 4

257. 0$ C 4 C 4.
Proof:/
h
# 4
i
L?

258. r

259. @ M! @ A B # ?

260. r5

261. @ M! @ 5 A BN
?

262. (r # r5 @ r r5 )

263. L @ M! @ N L @ M! @ 5 N
A B
R S S S T
S S S U
V W W W W W W W W X

264. k k k k k k l
j
@ M( @ r) #
@ M( @ r5)
@L @ M( @ r) N
L @ M( @ r5) N
p p p p p p q
o

265. ZL @ M! @ N
ZL @ M! @ 5 N
] ] ] ] ] ] ] ] ^
_ S S S `
S S S a
bc

266. @ M( @ r)

267. ZL @ M! @ N
d e

268. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
9
#bc

269. @ M( @ r5 )

270. ZL @ M! @ 5 N
d e
?

271. r

272. @ M! @ A B #
?

273. r5

274. @ M! @ 5 A B
#
4
Hence (a) is proved and similarly (b) is proved by analogy.
Proof:g
# 4
!EF

275. !

276. @ ! @ G H # EF

277. !5

278. @ ! @ 5G H
EF

279. ! # !5 @ !!5

280. ! @ ! @ ! @ ! @ 5G H
R S T
S U
V W W W W X

281. k k k l
j
@ ! @ !
# @ ! @ !5
)
@( @ ! @ !
) o
( @ ! @ !5
p p p q
! @ ! @ 5

282. ! @ ! @
] ] ] ] ^
_ S ` S a

283. ! @ ! @
F

284. @ ! @ !
G
B

285. ! @ ! @ 5
# F

286. @ ! @ !5
G
B
EF

287. !

288. @ ! @ G H #EF

289. !5

290. @ ! @ 5G H
# 4
Hence (c) is proved and similarly (d) is proved by analogy.
Theorem 3.4. For every IVIFS A and for every natural number + :
/$ h
i
f$
Proof./$
h
i
L

291. M
?

292. LM %
N

293. L @ M! @
%

294. @ M! @
NA BN

295. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
%

296. @ M! @
10
R S S T
S S U
V W W W W W X

297. @ Z @ M

298. Y @ Z @ M %
[

299. %
YZ @ M! @

300. Z @ M! @
[
] ] ] ] ] ^
_ S S `
S S a
R S S T
S S U
V W W W W W W X

301. k k k k l
j
x @ x @ Y @ Z @ M %
[y
y

302. [y
x @ x @ Y @ Z @ M
y
p p p p q
o

303. k k k l
j
%
YZ @ M! @
[

304. YZ @ M! @
[
p p p q
o
] ] ] ] ] ] ^
_ S S `
S S a
R S T
S U
z

305. lY @ YZ @ M %
j
[
[

306. Y @ YZ @ M
[
[
o
q

307. L @ M! @
N{
S a
_ S `
N

308. @ L @ M
78

309. m @ L @ M %
Nn

310. L @ M! @
%

311. @ M! @
N;

312. M
?

313. LM %
N

314. L @ M! @
%

315. @ M! @
NA B
Hence (a) is proved.
Proof.f$
!EF

316. !! %

317. !

318. ! @ ! @
%

319. @ ! @
G H
bc

320. ( @ ! @ ! %

321. @ ! @ !
)
%

322. (! @ ! @

323. ! @ ! @
)
d e

324. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
11
R S S S T
S S S U
V W W W W W W W X
k l

325. j
)i N

326. L @ Mh @ ( @ ! @ ! %
)i N
L @ Mh @ ( @ ! @ !
p q
o

327. %
YLM! @ ! @
N

328. LM! @ ! @
N[
] ] ] ] ] ] ] ^
S S S a
_ S S S `
R S T
S U
V W W W X
k l

329. j
@ LM! @ ! %
N

330. @ LM! @ !
p q
N
o
%

331. @ ! @

332. ! @ ! @
] ] ] ^
_ S `
S a
tu

333. ( @ ! @ ! %

334. @ ! @ !
)

335. ! @ ! @
%

336. @ ! @
v w
EF

337. !! %

338. !

339. ! @ ! @
%

340. @ ! @
G H
Hence (b) is proved.
Theorem 3.5: For every IVIFS A and for every natural number n+1:
(a). |h
|i
, (b). |!|
Proof. (a).
|L
|N
|h
%

341. EF

342. !

343. ! %
G}Hi

344. %

345. @ M @ !
|L?

346. L @ M @ !
N

347. LM! %

348. M!
NA
BN
|
k k k l
j
R S S T
S S U
V W W W W W X
%

349. Z @ M @ !

350. YZ @ M @ !
[

351. N
Y @ ZL @ M! %
N

352. @ ZL @ M!
[
] ] ] ] ] ^
_ S S `
S S a
p p p q
o

353. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.4, October 2014
)3 *
12
R S S T
S S U
V W W W W W X
N

354. Y @ ZL @ M! %
N

355. @ ZL @ M!
[

356. %

357. Z @ M @ !
YZ @ M @ !
[
] ] ] ] ] ^
_ S S `
S S a
Hence (a) is proved.
Proof. (b).
|!|
%

358. |!EF

359. !

360. ! %
G}H

361. %

362. @ ! @
|!EF

363. ! @ ! @

364. !! %
G H

365. !
%
|t2

366. (! @ ! @

367. ! @ ! @
)

368. ( @ ! @ ! %

369. @ ! @ !
)3
w
'2

370. ( @ ! @ ! %

371. @ ! @ !
)

372. (! @ ! @
%

373. ! @ ! @
Hence (b) is proved.
4. CONCLUSION
In this paper, two new operators based IVIFS were introduced and few theorems were proved. In
future, the application of this operator will be proposed and another two operators based on IVIFS
are to be introduced.
5. ACKNOWLEDGMENTS
The authors are highly grateful to the Editor-in-Chief and Reviewer Professor Ayad Ghany
Ismaeel and the referees for their valuable comments and suggestions.
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