Download Some properties of fuzzy real numbers

Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27
http://scma.maragheh.ac.ir
SOME PROPERTIES OF FUZZY REAL NUMBERS
BAYAZ DARABY1 AND JAVAD JAFARI2∗
Abstract. In the mathematical analysis, there are some theorems
and definitions that established for both real and fuzzy numbers.
In this study, we try to prove Bernoulli’s inequality in fuzzy real
numbers with some of its applications. Also, we prove two other
theorems in fuzzy real numbers which are proved before, for real
numbers.
1. Introduction
According to [5] if η is a non-negative fuzzy real number, then for
any real number p ̸= 0, η p is a fuzzy real number, wich some of their
properties were proved before. (This paper is trying to study some
properties wich are proved in [5] for two real fuzzy numbers η p and
δ p ). Also in [4], Bernoulli’s inequality states that, if α is a real number
whenever 1 + α ≥ 0, then we have (1 + α)n ≥ 1 + nα. Furthermore
(1 + α)n = 1 + nα if and only if α = 0 or n = 1, and this theorem
indicates for every positive x and positive integer n > 0, there is one
and only one positive real y such that y n = x (in this study we try to
prove the above theorems in fuzzy real numbers).
Section 2 contains basic definitions and theorems. In Section 3, we prove
Bernoulli’s inequality and two other theorems in fuzzy real numbers.
2. Preliminaries and basic facts
In this section, we introduce some preliminaries in the theory of fuzzy
real numbers. Let η be a fuzzy subset on R, i.e., a mapping η : R → [0, 1]
which associates with each real number t, its grade of membership η(t).
[3]
2010 Mathematics Subject Classification. 26E50, 03E72, 39B72.
Key words and phrases. Fuzzy real number, Bernoulli’s inequality, Real number.
Received: 9 August 2015, Accepted: 30 November 2015.
∗
Corresponding author.
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B. DARABY AND J. JAFARI
Definition 2.1 ([2]). A fuzzy subset η on R is called a fuzzy real number, whose its α-level set is denoted by [η]α , i.e., [η]α = {t : η(t) ≥ α},
if it satisfies two axioms:
(N1 ) There exists t0 ∈ R such that η(t0 ) = 1.
(N2 ) For each α ∈ (0, 1], [η]α = [ηα1 , ηα2 ] where −∞ < ηα1 ≤ ηα2 < +∞.
The set of all fuzzy real numbers denoted by F (R). If η ∈ F (R) and
η(t) = 0 whenever t < 0, then η is called a non-negative fuzzy real number and F ∗ (R) denotes the set of all non-negative fuzzy real numbers.
The number 0 stands for the fuzzy real number as:
{
1,
t = 0,
0(t) =
0,
t ̸= 0.
Clearly, 0 ∈ F ∗ (R). Also the set of all real numbers can be embedded in
F (R), because, if r ∈ (−∞, ∞), then r ∈ F (R) satisfies r(t) = 0(t − r).
Theorem 2.2 ([1]). Let {[aα , bα ] : α ∈ (0, 1]} be a family of nested bounded
closed intervals and η : R → [0, 1] be a function defined by
η(t) = sup{α ∈ (0, 1] : t ∈ [aα , bα ]}.
Then η is a fuzzy real number.
Theorem 2.3 ([1]). Let [aα , bα ], 0 < α ≤ 1, be a family of non-empty
intervals. If
(i) [aα1 , bα1 ] ⊃ [aα2 , bα2 ] for all 0 < α1 ≤ α2 ≤ 1,
(ii) [ lim aαk , lim bαk ] = [aα , bα ] whenever {αk } is an increasing
k→∞
k→∞
sequence in (0, 1] converging to α,
then the family [aα , bα ] represents the α-level sets of a fuzzy real number
η such that η(t) = sup{α ∈ (0, 1] : t ∈ [aα , bα ]} and [η]α = [ηα2 , ηα2 ] =
[aα , bα ].
Definition 2.4 ([2]). Fuzzy arithmetic operations ⊕, ⊖, ⊗ and ⊘ on
F (R) × F (R) can be defined as:
(i) (η ⊕ δ)(t) = sups∈R {η(s) ∧ δ(t − s)}, t ∈ R,
(ii) (η ⊖ δ)(t) = sups∈R {η(s) ∧ δ(s − t)}, t ∈ R,
(iii) (η ⊗ δ)(t) = sups∈R,s̸=0 {η(s) ∧ δ(t/s)}, t ∈ R,
(iv) (η ⊘ δ)(t) = sups∈R {η(st) ∧ δ(s)}, t ∈ R.
Definition 2.5 ([2]). For k ∈ R \ 0, fuzzy scalar multiplication k ⊙ η is
defined as (k ⊙ η) = η(t/k) and 0 ⊙ η is defined to be 0.
SOME PROPERTIES OF FUZZY REAL NUMBERS
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Lemma 2.6 ([5]). Let η and δ be fuzzy real numbers. Then
∀t ∈ R,
η(t) = δ(t)
⇔
∀α ∈ (0, 1],
[η]α = [δ]α .
Lemma 2.7 ([2]). Let η, δ ∈ F (R) and [η]α = [ηα1 , ηα2 ] and [δ]α =
[δα1 , δα2 ]. Then
(i) [η ⊕ δ]α = [ηα1 + δα1 , ηα2 + δα2 ],
(ii) [η ⊖ δ]α = [ηα1 − δα2 , ηα2 − δα1 ],
(iii) [η ⊗ δ]α = [ηα1 δα1 , ηα2 δα2 ], η, δ ∈ F ∗ (R),
(iv) [1 ⊘ δ]α = [1/δα2 , 1/δα1 ], δα1 > 0.
Definition 2.8 ([5]). Let η be a non-negative fuzzy real number and
p ̸= 0 be a real number. We define η p as:
{
η(t1/p ),
t ≥ 0,
η p (t) =
p
η (t) = 0,
t < 0.
Also we set η p = 1, in the case p = 0.
In [5], Sadeqi and al. showed that η p is a non-negative fuzzy real number,
i.e., η p ∈ F ∗ (R), ∀p ∈ R. We need to investigate conditions (N1 ) and
(N2 ) in the definition of fuzzy real numbers. For condition (N1 ), since
η is a fuzzy real number, there exists t0 ∈ [0, ∞), such that η(t0 ) = 1.
Set t′ =tp0 . Then
η p (t′ ) = η((t′ )1/p ) = η((tp0 )1/p ) = η(t0 ) = 1.
For condition (N2 ), since [η]α = [ηα1 , ηα2 ], for all α ∈ (0, 1], we have
[η p ]α = {t; η p (t) ≥ α}
= {t; η(t1/p ) ≥ α}
= {sp ; η(s) ≥ α}
= ({s; η(s) ≥ α})p
= ([ηα1 , ηα2 ])p .
Therefore, η p is a fuzzy real number. Also it is clear that if p > 0, then
[η p ]α = [(ηα1 )p , (ηα2 )p ] and if p < 0, then [η p ]α = [(ηα2 )p , (ηα1 )p ].
Theorem 2.9 ([5]). Let η be a non-negative fuzzy real number and p, q
be non-zero integers. Then
(i) p > 0 ⇒ η p = ⊗pi=1 η,
(ii) p < 0 ⇒ η p = 1 ⊘ (⊗−p
i=1 η),
(iii) η p ⊗ η q = η p+q ; pq > 0,
(iv) (η p )q = η pq .
Definition 2.10 ([2]). Define a partial ordering ⪯ in F (R) by η ⪯ δ if
and only if ηα1 ≤ δα1 and ηα2 ≤ δα2 for all α ∈ (0, 1]. The strict inequality
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B. DARABY AND J. JAFARI
in F (R) is defined by η ≺ δ if and only if ηα1 < δα1 and ηα2 < δα2 for all
α ∈ (0, 1].
3. Main results
Using results of the last section and an idea of W. Rudin [4], we prove
the following theorems.
Proposition 3.1. Let η, δ be fuzzy real numbers and p be a non-zero
integer. Then
(i) η p ⊗ δ p = (η ⊗ δ)p = ⊗pi=1 (η ⊗ δ) if p > 0,
(ii) η p ⊗ δ p = 1 ⊘ (⊗−p
if p < 0,
i=1 (η ⊗ δ))
(iii) η p ⊘ δ p = (η ⊘ δ)p = ⊗pi=1 (η ⊘ δ).
Proof. We prove this proposition by using Lemmas 2.6, 2.7 and Theorem 2.9
(i) Let p > 0 and α ∈ (0, 1]. Then
(3.1)
[
]
[η p ⊗ δ p ]α = (ηα1 )p (δα1 )p , (ηα2 )p (δα2 )p
[ p
]
p
∏
∏
=
(ηα1 δα1 ), (ηα2 δα2 )
i=1
i=1
= [⊗pi=1 η ⊗ δ]α .
(ii) For all α ∈ (0, 1] and p < 0, we have
(3.2)
(
)p
[η p ⊗ δ p ]α = [ηα1 δα1 , ηα2 δα2 ]
[
]
= (ηα2 δα2 )p , (ηα1 δα1 )p
[
]
= 1/(ηα2 δα2 )−p , 1/(ηα1 δα1 )−p
[ −p
]
−p
∏
∏
= 1/ (ηα2 δα2 ), 1/ (ηα1 δα1 )
[
i=1
= 1⊘
(⊗−p
i=1 η
i=1
]
⊗ δ) .
α
SOME PROPERTIES OF FUZZY REAL NUMBERS
25
(iii) Let p > 0 and α ∈ (0, 1]. Then
(3.3)
[
]
[η p ⊘ δ p ]α = (ηα1 /δα2 )p , (ηα2 /δα1 )p
[ p
]
p
∏
∏
(ηα1 /δα2 ), (ηα2 /δα1 )
=
i=1
i=1
= [⊗pi=1 (η ⊘ δ)]α .
In case p < 0, the proof is similar. It is easy to check the other
cases.
□
Theorem 3.2. For any non-negative fuzzy real number η > 0 and any
integer n > 0, there is one and only one fuzzy real number δ > 0 such
that δ n = η.
Proof. Let δ be a non-negative fuzzy real number. Then, according to
Definition 2.8, δ n is a fuzzy real number and also, according to the Theorem 2.9, [δ n ]α = [(δα1 )n , (δα2 )n ], since both δα1 and δα2 are non-negative
real numbers. Using [4], we have (δα1 )n = aα and (δα2 )n = bα . Therefore,
we have [δ n ]α = [aα , bα ] = [η]α . Now, let δ1 , δ2 ∈ F ∗ (R), (δ1 )n = η and
(δ2 )n = η, then δ1 = δ2 . Hence δ is a unique fuzzy real number.
□
Corollary 3.3. If a and b are two positive fuzzy real numbers and n is
a positive integer, then (a ⊗ b)1/n = (a1/n ) ⊗ (b1/n ).
Proof. Put η = a1/n and δ = b1/n . According to Proposition 3.1 we have
a ⊗ b = η n ⊗ δ n = (η ⊗ δ)n , so, according to Theorem 3.2, we have:
(a ⊗ b)1/n = η ⊗ δ
= (a1/n ) ⊗ (b1/n ).
□
Theorem 3.4. (Bernoulli’s inequality) Let α be a fuzzy real number
such that 1 ⊕ α ⪰ 0. Then, for all n ∈ N , we have (1 ⊕ α)n ⪰ 1 ⊕ nα.
Furthermore (1 ⊕ α)n = 1 ⊕ nα ⇔ α = 0 or n = 1.
Proof. We prove this theorem by using Definition 2.10 and Theorem 2.9.
Using mathematical induction, for n = 1, we have (1 ⊕ α) ⪰ 1 ⊕ α. If
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B. DARABY AND J. JAFARI
the theorem is true for n = k, then for n = k + 1 we have
(3.4)
(1 ⊕ α)k+1 = (1 ⊕ α)k ⊗ (1 ⊕ α)
⪰ (1 ⊕ kα) ⊗ (1 ⊕ α)
= 1 ⊕ α ⊕ kα ⊕ kα2
⪰ 1 ⊕ α ⊕ kα
= 1 ⊕ (1 + k)α,
hence, the theorem is true for n = k + 1 too.
If α = 0 or n = 1, the proof is clear. Conversely, let (1 ⊕ nα) = (1 ⊕ α)n
and n ̸= 0, then we have 1 ⊕ nα = (1 ⊕ α)n ⪰ 1 ⊕ nα ⊕ (n − 1)α2 and
hence
1 ⪰ 1 ⊕ (n − 1)α2
⇒
(n − 1)α2 ⪯ 0,
we know that n − 1 > 0, so we conclude that α2 ⪯ 0. It follows that
α2 = 0, hence α = 0. Then equality is established.
□
Corollary 3.5. Let β be
√ a fuzzy real number such that√1 ⊕ β ⪰ 0. Then
for all n ∈ N, we have n 1 ⊕ β ⪯ 1 ⊕ β/n. Therefore n 1 ⊕ β= 1 ⊕ β/n
if and only if β = 0 or n = 1.
Proof. Let β√
⊘n = α. Using of Bernoulli’s
inequality, we have (1⊕α)n ⪰
√
n
□
1 ⊕ nα. So 1 ⊕ nα ⪯ 1 ⊕ α or n 1 ⊕ β ⪯ 1 ⊕ β/n.
Theorem 3.6. Let x, y be fuzzy real numbers and x ⊕ y ⪰ 0. Then for
all n ∈ N, we have
(x ⊕ y/2)n ⪯ xn ⊕ y n /2.
Proof. We prove this theorem by using Definition 2.10 and Theorem 2.9.
Using mathematical induction, for n = 1, we have (x ⊕ y/2) ≤ x ⊕ y/2.
If theorem is true for n = k, then for n = k + 1 we have
(3.5)
(x1α + yα1 /2)k+1 ≤ (x1α )k + (yα1 )k /2 × (x1α + yα1 /2)
= (x1α )k+1 + (x1α )k yα1 + x1α (yα1 )k + (yα1 )k+1 /4
≤ (x1α )k+1 + (yα1 )k+1 /2.
Similarly we have (x2α +yα2 /2)k+1 ≤ (x2α )k+1 +(yα2 )k+1 /2. Hence from the
above inequality and Definition 2.10, we have (x⊕y/2)n ⪯ xn ⊕y n /2. □
SOME PROPERTIES OF FUZZY REAL NUMBERS
27
References
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3. M. Mizumoto and K. Tanaka, Some properties of fuzzy numbers, in: M.M. Gupta
et al., Eds., Advances in Fuzzy Set Theory and Applications (North-Holland,
New York, 1979), 153–164.
4. W. Rudin, Principles of Mathetical Analysis, Mcgraw-Hill, New York, 1976.
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1
Department of Mathematics, Faculty of Science, University of Maragheh,
Maragheh, Iran.
E-mail address: [email protected]
2
Department of Mathematics, Faculty of Science, University of Maragheh,
Maragheh, Iran.
E-mail address: [email protected]