Appl. Math. J. Chinese Univ.
2014, 29(3): 253-264
On uniqueness of (fuzzy) relation in some generalized
rough set model
PEI Dao-wu
Abstract. In rough set theory, crisp and/or fuzzy binary relations play an important role in
both constructive and axiomatic considerations of various generalized rough sets. This paper
considers the uniqueness problem of the (fuzzy) relation in some generalized rough set model.
Our results show that by using the axiomatic approach, the (fuzzy) relation determined by
(fuzzy) approximation operators is unique in some (fuzzy) double-universe model.
§1
Introduction
As meaningful generalizations of rough sets proposed by Pawlak [6], various generalized
rough sets have become important tools in representing and managing uncertainty contained
in information systems. These models include the rough set model based on a general binary
relation, called the relation based model [26-27], the rough set model based on a general fuzzy
binary relation, called the fuzzy relation based model which takes rough fuzzy sets and fuzzy
rough sets as its special cases [4, 25], the rough set model based on a covering of the universe,
called the covering based model [29-30].
There are two main approaches in studying the (fuzzy) relation based rough sets: the constructive approach and the axiomatic approach (or the algebraic approach). The former has
been widely used since Pawlak [13] was published in 1982. Based on the approach, one constructs a (fuzzy) approximation space and two (fuzzy) approximation operators from a (fuzzy)
binary relation on some universe, and then investigates properties of the (fuzzy) approximation
operators. And the latter was used by Lin and Liu [10] in 1994 for the first time. Based on
the approach, one proposes some suitable conditions, called axioms, which are satisfied by two
(fuzzy) set-valued operators on some universe, then considers the existence of a (fuzzy) binary
relation, and the (fuzzy) relation is used to construct a (fuzzy) approximation space such that
Received: 2011-03-30.
MR Subject Classification: 03E72, 94D05.
Keywords: fuzzy relation, rough set, fuzzy rough set, lower approximation operator, upper approximation
operator.
Digital Object Identifier(DOI): 10.1007/s11766-014-2798-x.
Supported by the National Natural Science Foundation of China (11171308, 61379018, 51305400).
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Vol. 29, No. 3
in this space, the two operators equal to the approximation operators respectively [10-12, 14-15,
17, 19-20, 22-24, 26, 29].
On the literature of the axiomatic considerations of rough set theory, almost all authors
discussed the existence problem of the (fuzzy) relations determined by (fuzzy) approximation
operators. For example, Lin and Liu [10], Yao [26-27], Wu, et al. [22-24], Mi and Zhang [11]
only considered the existence in the corresponding rough set models. Thiele [17-20] presented
some results concerning the correspondence between all (fuzzy) relations and all (fuzzy) approximation operators based on suitably defined bijections. And Morsi, et al. [12] presented
similar results to fuzzy rough set model induced by some special t-norm. However, it is very
strange that few author considered the uniqueness problem of the (fuzzy) relations in the literature. In fact, if the uniqueness holds in some (fuzzy) rough set model, then we can obtain
more concise conclusions to some problems in rough set theory. For example, the model will
be more simple, the discussion will be more easy, and the uncertainty contained in the system
will be significantly minimized. Conversely, if the uniqueness does not hold in some (fuzzy)
rough set model, we should consider selecting suitable one to optimize the model, or consider
relationships between these models. Therefore, the uniqueness problem of the (fuzzy) relation
in some (fuzzy) rough set model is both interesting and important.
This paper considers the uniqueness of the (fuzzy) relation in some (fuzzy) rough set model
based on two universes. Some new results with respect to these models are presented.
§2
Uniqueness of crisp relation in relation based model
This section considers the uniqueness problem of the crisp binary relation determined by
rough approximation operators in a generalized rough set model based on double universes [15,
21, 28].
Let U and V be two finite universes. P(U ) and P(V ) denote the power sets of U and V
respectively. A subset R of U × V is called a (crisp) binary relation from U to V . For any
x ∈ U , denote
Rs (x) = {y ∈ V | (x, y) ∈ R}, Rp (x) = {y ∈ V | (y, x) ∈ R}.
R is serial if for any x ∈ U , there exists a y ∈ U such that (x, y) ∈ R, i.e., Rs (x) = ∅; R
is inverse serial if for any x ∈ U , there exists a y ∈ U such that (y, x) ∈ R, i.e., Rp (x) = ∅.
In the case U = V , R is reflexive if for any x ∈ U , we have (x, x) ∈ R; R is symmetric if for
all x, y ∈ U , (x, y) ∈ R implies (y, x) ∈ R; R is transitive if for all x, y, z ∈ U , (x, y) ∈ R and
(y, z) ∈ R imply (x, z) ∈ R; R is Euclidean if for all x, y, z ∈ U , (x, y) ∈ R and (x, z) ∈ R imply
(y, z) ∈ R.
An equivalence relation on U is a reflexive, symmetric and transitive relation on U .
Definition 2.1. [28] Let U and V be two universes, and R be a compatibility relation from U
to V . The ordered triple (U, V, R) is called a (two-universe) approximation space. The lower
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On uniqueness of (fuzzy) relation in some generalized rough set model
255
and upper approximations of A ⊆ V are respectively defined as follows:
aprR (A)(x)
aprR (A)(x)
=
=
{x ∈ U |Rs (x) ⊆ A},
{x ∈ U |Rs (x) ∩ A = ∅}.
(1)
The ordered set-pair (aprR (A), apr R (A)) is called a generalized rough set. Moreover, a
subset A of V is said to be definable if aprR (Y ) = aprR (Y ), otherwise it is undefinable, or
rough.
Theorem 2.1. [28] In a two-universe model (U, V, R), the approximation operators L = aprR
and H = aprR satisfy the following properties for any A, B ⊆ V :
(L1) L(A) =∼ H(∼ A);
(U1) H(A) =∼ L(∼ A);
(L2) L(V ) = U ;
(U2) H(∅) = ∅;
(L3) L(A ∩ B) = L(A) ∩ L(B);
(U3) H(A ∪ B) = H(A) ∪ H(B);
(L4) L(A ∪ B) ⊇ L(A) ∩ L(B);
(U4) H(A ∩ B) ⊆ H(A) ∩ H(B);
(L5) A ⊆ B =⇒ L(A) ⊆ L(B);
(U5) A ⊆ B =⇒ H(A) ⊆ H(B);
(L6) L(∅) = ∅;
(U6) H(V ) = U .
It is obvious that lower and upper approximations are special interior and closure operators
respectively [1-3].
Remark 2.1. We observe the fact that in the above model, the two approximations of a subset
of the universe V are subsets of the universe U . This is not too reasonable. Pei and Xu [15]
gave a revised model in which the two approximations of a subset of V are still subsets of V .
Theorem 2.2. [15] Suppose that R is a binary relation from U to V . Denote L = apr R ,
H = aprR , then the following conditions give characterization to several important relations:
(i) R is serial if and only if one of the following conditions holds:
(L7) L(A) ⊆ H(A) for any A ⊆ V ;
(U7) H(V ) = U ;
(LU7) L(∅) = ∅.
(ii) R is inverse serial if and only if one of the following conditions holds:
(L8) L(A) = U for any A ⊂ U ;
(U8) H(A) = ∅ for any ∅ = A ⊆ U ;
(LU8) Rs (U ) = V ;
(U8’) H(y) = ∅ for any y ∈ V .
For the generalized rough set models given by Definition 2.1, we have the following algebraic
characterization of two approximation operators based on the so-called axiomatic approach or
algebraic approach.
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Theorem 2.3. Let U and V be two universes, and L and H two dual operators from P(V )
to P(U ), i.e.,
L(A) =∼ H(∼ A), H(A) =∼ L(∼ A), A ⊆ V.
(2)
If L satisfies the conditions (L1) and (L2), or equivalently, the operator H satisfies the conditions (U1) and (U2), then there exists a binary relation R ⊆ U × V such that in the approximation space (U, V, R), the operators L and H are lower and upper approximation operators
respectively, more precisely,
L(A) = aprR (A), H(A) = aprR (A), A ⊆ V.
(3)
Proof. “=⇒”. The necessity follows immediately from Theorem 2.2.
“⇐=”. Suppose that the operator H satisfies the conditions (U1) and (U2). Define R ⊆
U × V as follows:
(x, y) ∈ R ⇐⇒ x ∈ H({y}), x ∈ U, y ∈ V.
Thus
Rs (x) = {y ∈ V | x ∈ H({y})},
or equivalently,
H({y}) = {x ∈ U | y ∈ Rs (x)}.
By the definition of R, we have
aprR (∅) = ∅ = H(∅),
apr R ({y}) = {x ∈ U | ∃t ∈ Rs (x), t ∈ {y}}
= {x ∈ U | y ∈ Rs (x)}
= H({y}), y ∈ V.
Moreover, by the condition (U2) and the finiteness of the universe U , we have
aprR (A) =
aprR ({y})
y∈A
=
y∈A H({y})
= H(A), A ⊆ V.
Finally, by the duality of two operators H and L, we have
aprR (A) = L(A), A ⊆ V.
And this completes the proof of the theorem.
We can similarly prove the existence of the serial (or reflexive, symmetric, transitive, Euclidean) relation determined by the corresponding operators.
For convenience, we call an operator L satisfying the conditions (L1) and (L2) an LA
operator (for lower approximation operator), and an operator H satisfying the conditions (U1)
and (U2) a UA operator (for upper approximation operator).
According to Theorem 2.3, there is a binary relation R from U to V . Naturally, the following
question arises:
Problem 1. Whether is the binary relation determined by Theorem 2.3 unique?
In order to answer the question, we first present the following two theorems.
Theorem 2.4. Suppose that R1 , R2 ⊆ U ×V , then the following four conditions are equivalent:
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On uniqueness of (fuzzy) relation in some generalized rough set model
257
(i) R1 ⊆ R2 ;
(ii) (R1 )s (x) ⊆ (R2 )s (x), x ∈ U ;
(iii) aprR (A) ⊇ apr R (A), A ⊆ V ;
1
2
(iv) aprR1 (A) ⊆ aprR2 (A), A ⊆ V .
Proof. (i)=⇒(ii). Suppose that (i) holds, i.e., R1 ⊆ R2 . Then for any x ∈ U , we have
(R1 )s (x)
= {y ∈ V | (x, y) ∈ R1 }
⊆ {y ∈ V | (x, y) ∈ R2 }
= (R2 )s (x).
(ii)=⇒(iii). Suppose that (ii) holds, i.e., (R1 )s ⊆ (R2 )s . Then for any A ⊆ V , we have
apr R (A)
1
=
{x ∈ U | (R1 )s (x) ⊆ A}
⊇
=
{x ∈ U | (R1 )s (x) ⊆ A}
aprR (A).
2
(iii)=⇒(iv). If (iii) holds, then we have
apr R1 (A)
=
⊆
=
∼ apr R (∼ A)
1
∼ apr R (∼ A)
2
aprR2 (A).
(iv)=⇒(i). Suppose that (iv) holds, i.e., for any A ⊆ V , apr R1 (A) ⊆ aprR2 (A). If (i) does
not hold, i.e.,
R1 ⊆ R2 ,
or equivalently, there is (x, y) ∈ R1 − R2 such that
y ∈ (R1 )s(x), y ∈ (R2 )s(x),
then we have
x ∈ aprR1 ({y}), x ∈ aprR2 ({y}).
This contracts to the assumption (iv).
Theorem 2.5. Suppose that R1 , R2 ⊆ U ×U , then the following four conditions are equivalent:
(i) R1 = R2 ;
(ii) (R1 )s (x) = (R2 )s (x), x ∈ U ;
(iii) aprR (A) = apr R (A), A ⊆ V ;
1
2
(iv) aprR1 (A) = aprR2 (A), A ⊆ V .
Proof. This theorem follows by using Theorem 2.4 two times.
The following theorem gives a positive answer to Problem 1:
Theorem 2.6. The binary relation R in Theorem 2.3 is unique, and it is defined as follows:
(x, y) ∈ R ⇐⇒ x ∈ H({y}), x, y ∈ U × V.
(4)
Proof. In fact, if there is another relation Q from U to V such that (3) holds, i.e., L(A) =
apr Q (A), H(A) = aprQ (A), for any A ⊆ V , then we have
apr R (A) = L(A) = aprQ (A).
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Thus we have R = Q by Theorem 2.5. Finally, (4) holds obviously by the proof of Theorem 2.3
and the proved uniqueness of the relation.
As a natural corollary of the above theorem, we can prove the uniqueness of the serial (or
reflexive, symmetric, transitive, Euclidean) relation determined by the corresponding operators.
Remark 2.2. According to Theorem 2.6, there is only one correspondence between all LA
operators (or all UA operators) and all binary relations on U in the Thiele’s sense [17].
§3
Uniqueness of fuzzy relation in fuzzy rough set model
This section considers the uniqueness of the fuzzy binary relation determined by fuzzy rough
approximation operators in a fuzzy rough set model based on two universes.
Let U be a universe. Here we do not restrict the universe to be finite. A mapping A : U →
[0, 1] is called a fuzzy set on U . We use F (U ) to denote the set of all fuzzy sets on U .
Fuzzy sets λu , α̂ with the following forms are called fuzzy points and fuzzy constants respectively:
1, x = u;
λu (x) =
0, x = u.
α̂(x) = α, x ∈ U.
For more details of fuzzy mathematics, we refer to Klir and Yuan [9].
A t-norm is an increasing, associative and commutative mapping T : [0, 1]2 → [0, 1] satisfying the boundary condition T (1, x) = x for all x ∈ [0, 1].
A fuzzy implication is an extension of the classical implication on [0,1], i.e., a mapping
I : [0, 1]2 → [0, 1] satisfying the following conditions
I(1, 0) = 0, I(0, 0) = I(0, 1) = I(1, 1) = 1.
(5)
Let U and V be two universes. A fuzzy subset R of U × V is called a fuzzy binary relation
from U to V . R is serial if for all x ∈ U , we have y∈V R(x, y) = 1; R is inverse serial if for
any y ∈ V , we have x∈U R(x, y) = 1. In the case of U = V , R is reflexive if R(x, x) = 1 holds
for any x ∈ U ; R is symmetric if R(x, y) = R(y, x) holds for all x, y ∈ U . R is T -transitive
if T (R(x, y), R(y, z)) ≤ R(x, z) holds for all x, y, z ∈ U . Particularly, R is transitive if R is
min-transitive.
A fuzzy relation R is called a fuzzy T -equivalence relation on U if it is reflexive, symmetric
and T -transitive. Particularly, a fuzzy relation R is called a fuzzy equivalence relation on U if
it is a min-equivalence relation on U .
Definition 3.1. [25] Let U and V be two universes, and R be a fuzzy relation from U to V .
The ordered triple (U, V, R) is called a (double-universe) fuzzy approximation space. The lower
and upper approximations of A ∈ F(V ) are defined respectively as follows:
{A(y) ∨ (1 − R(x, y)) | y ∈ V },
Apr R (A)(x) =
(6)
Apr R (A)(x) =
{A(y) ∧ R(x, y) | y ∈ V }.
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On uniqueness of (fuzzy) relation in some generalized rough set model
259
The ordered fuzzy set-pair (Apr R (A), Apr R (A)) is called a fuzzy rough set. Moreover, A is
definable if
AprR (Y ) = AprR (Y ),
otherwise it is undefinable, or rough.
It is well known that the fuzzy rough set model defined above is an extension of the rough
fuzzy set model and the fuzzy rough set model defined by Dubois and Prade [4] or [14, 23-25].
Theorem 3.1. [24] In a double-universe fuzzy model (U, V, R), the fuzzy approximation operators L = Apr R and H = Apr R satisfy the following conditions for any A, B ∈ F(V ) and
α ∈ [0, 1]:
(F L1) L(A) =∼ H(∼ A);
(F U 1) H(A) =∼ L(∼ A);
(F L2) L(A ∪ α
) = L(A) ∪ α
;
(F U 2) H(A ∩ α
) = H(A) ∩ α
;
(F L3) L(A ∩ B) = L(A) ∩ L(B);
(F U 3) H(A ∪ B) = H(A) ∪ H(B);
(F L4) A ⊆ B =⇒ L(A) ⊆ L(B);
(F U 4) A ⊆ B =⇒ H(A) ⊆ H(B);
(F L5) L(A ∪ B) ⊇ L(A) ∩ L(B);
(F U 5) H(A ∩ B) ⊆ H(A) ∩ H(B).
Theorem 3.2. [24] Let R be a fuzzy binary relation from U to V , L = Apr R , and H = Apr R .
Then
(i) R is serial if and only if one of the following conditions holds:
(FL6) L(∅) = ∅,
(FU6) H(V ) = U ,
(FLU6) L(A) ⊆ H(A) for any A ∈ F(V ),
(FL6’) L(
α) = α
for all α ∈ [0, 1],
(FU6’) H(
α) = α
for all α ∈ [0, 1];
(ii) R is reflexive if and only if one of the following conditions holds:
(F L7) L(A) ⊆ A,
(F U 7) A ⊆ H(A);
(iii) R is symmetric if and only if one of the following conditions holds:
(F L8) L(λU\{y} )(x) = L(λU\{x} )(y),
(F U 8) H(λx )(y) = H(λy )(x),
(F L9) A ⊆ L(H(A)),
(F U 9) H(L(A)) ⊆ A;
(iv) R is transitive if and only if one of the following conditions holds:
(F L10) L(A) ⊆ L(L(A)),
(F U 10) H(H(A)) ⊆ H(A).
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For the fuzzy rough set model given by Definition 3.1, we have the following algebraic
characterization of the fuzzy approximation operators:
Theorem 3.3. Let U and V be two universes, and L and H be two dual operators from F (V )
to F (U ), i.e.,
L(A) =∼ H(∼ A), H(A) =∼ L(∼ A), A ∈ F(V ).
(7)
If L satisfies the conditions (FL1) and (FL2), or equivalently, the operator H satisfies the
conditions (FU1) and (FU2) for all A, B ∈ F(V ) and α ∈ [0, 1], then there exists a fuzzy
relation R ∈ F(U ×V ) such that in the fuzzy approximation space (U, V, R), the operators L and
H are the fuzzy lower and upper approximation operators respectively, i.e., for any A ∈ F(V ),
L(A) = Apr R (A), H(A) = Apr R (A).
(8)
Suppose that R is a fuzzy relation from U to V . Set
FR : U → F (V ), x → FR (x),
where
FR (x)(y) = R(x, y), y ∈ V.
For convenience, we call an operator L satisfying the conditions (FL1) and (FL2) an FLA
operator, and an operator H satisfying the conditions (FU1) and (FU2) an FUA operator.
For the fuzzy rough set case, we also have the uniqueness problem of the fuzzy relation.
According to Theorem 3.3, there is a fuzzy binary relation R from U to V . Naturally, the
following question arises:
Problem 2. Whether is the fuzzy relation determined by Theorem 3.3 unique?
Theorem 3.4. Suppose that R1 , R2 ∈ F(U × V ), then the following four conditions are
equivalent:
(i) R1 ⊆ R2 ;
(ii) FR1 (x) ⊆ FR2 (x), x ∈ U ;
(iii) Apr R (A) ⊇ AprR (A), A ∈ F(V );
1
2
(iv) Apr R1 (A) ⊆ Apr R2 (A), A ∈ F(V ).
Proof. (i)⇐⇒(ii). This is obvious from the definition of F .
(i)=⇒(iii). Suppose that (i) holds, i.e., R1 ⊆ R2 . Then for any A ∈ F(V ) and x ∈ U , we
have
Apr R (A)(x) =
y∈V [(1 − R1 (x, y)) ∨ A(x)]
1
≥
y∈V [(1 − R2 (x, y)) ∨ A(x)]
= AprR (A)(x).
2
(iii)=⇒(iv). This can be proved by the duality of two approximation operators.
(iv)=⇒(i). Suppose that (iv) holds, i.e., AprR1 ⊆ AprR2 . If (i) does not hold, i.e., R1 ⊆ R2 ,
then there is (x, y) ∈ U × V such that R1 (x, y) > R2 (x, y). Thus we obtain
Apr R1 ({y})(x) =
>
=
R1 (x, y)
R2 (x, y)
AprR2 ({y})(x).
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261
This contrasts to the assumption (iv).
If we use the above theorem two times, then we can obtain the following theorem.
Theorem 3.5. Suppose that R1 , R2 ∈ F(U × V ), then the following four conditions are
equivalent:
(i) R1 = R2 ;
(ii) FR1 = FR2 ;
(iii) Apr R = Apr R ;
1
2
(iv) Apr R1 = Apr R2 .
Moveover, we can derive the uniqueness of the fuzzy relation from Theorem 3.5, and also
give a positive answer to Problem 2.
Theorem 3.6. The fuzzy binary relation R in Theorem 3.3 is unique, and it is defined as
follows:
(9)
R(x, y) = H(λy )(x), (x, y) ∈ U × V.
In fact, we can similarly prove this theorem as we did in the proof of Theorem 2.6 in the
crisp case. For (9) we refer to [24].
Remark 3.1. According to Theorem 3.6, there is only one correspondence between all FLA
operators (or all FUA operators) and all fuzzy binary relations on U in the Thiele’s sense [18,
19].
As applications of the main results given above, we consider the so-called (T, I) type fuzzy
rough sets. For details, we refer to [16] and [22].
Definition 3.2. [16] Let U and V be two universe, R a fuzzy relation from U to V , T a t-norm,
and I a fuzzy implication. The triple (U, V, R) is called a (T, I) type (fuzzy) approximation
space. For A ∈ F(V ), the I-lower and T -upper approximations of A with respect to (U, V, R)
are defined as fuzzy sets Apr I (A) and Apr T (A) respectively, where
Apr IR (A)(x) =
{I(R(x, y), A(y)) | y ∈ V },
T
Apr R (A)(x) =
{T (R(x, y), A(y)) | y ∈ V }.
T
The couple (Apr IR (A), Apr R (A)) is called a (T, I) type fuzzy rough set.
An operator H : F (V ) → F (U ) is called a T -upper approximation operator if for all
Aj ∈ F(V ) (j ∈ J), A ∈ F(V ) and α ∈ [0, 1],
(TFU1) H(
α ∩T A) = α
∩T H(A),
(TFU2) H( j∈J Aj ) = j∈J H(Aj ),
where the operation ∩T is defined as follows:
(A ∩T B)(x) = T (A(x), B(x)), A, B ∈ F(V ), x ∈ V.
Theorem 3.7. [22] An operator H : F (V ) → F (U ) is a T -upper approximation operator if
and only if there is a fuzzy relation R ∈ F(U × V ) such that
T
H = Apr R .
(10)
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Theorem 3.8. The fuzzy binary relation R in Theorem 3.7 is unique, and it is defined as
follows:
(11)
R(x, y) = H(λy )(x), (x, y) ∈ U × V.
Proof. In fact, if there is another fuzzy relation Q from U to V such that (10) holds, i.e.,
T
H(A) = AprQ (A) for any A ⊆ V , then we have
T
T
Apr R (A) = H(A) = Apr Q (A).
Thus we have R = Q by Theorem 3.5.
An operator L : F (V ) → F (U ) is called an I-lower approximation operator if for all
Aj ∈ F(V ) (j ∈ J), A ∈ F(V ) and α ∈ [0, 1],
→I L(A),
(TFL1) L(
α →I A) = α
(TFL2) L( j∈J Aj ) = j∈J L(Aj ),
where the operation →I is defined as follows:
(A →I B)(x) = I(A(x), B(x)), A, B ∈ F(V ), x ∈ V.
Theorem 3.9. [15] Suppose that the fuzzy implication I satisfies the following condition:
I(a, I(b, c)) = I(b, I(a, c)), a, b, c ∈ [0, 1],
then an operator L : F (V ) → F (U ) is an I-lower approximation operator if and only if there
is a fuzzy relation R ∈ F(U × V ) such that
L = Apr IR .
(12)
Similarly to Theorem 3.8, we can easily obtain the following result by applying Theorem
3.5:
Theorem 3.10. The fuzzy binary relation R in Theorem 3.9 is unique, and it is defined as
follows:
(13)
R(x, y) = L(χV \{y} )(x), (x, y) ∈ U × V,
where χV \{y} is the characteristic function of the crisp set V \ {y}.
Thus we proved the uniqueness of the fuzzy relation in the (T, I) type fuzzy rough set model
given by Definition 3.2.
Remark 3.2. In fuzzy logic, an important concept related to t-norms is the residual implications [5-8]. It is easy to see that in Theorem 3.9 if I is a residual implication then the conclusion
is true.
§4
Conclusions
In the rough set theory, (fuzzy) binary relations play an important role in constructive and
axiomatic characterization of a (fuzzy) relation based rough set model. This paper investigated
the uniqueness of the (fuzzy) relation in the axiomatic consideration of rough set theory on two
universes. According to the results given in the present paper, based on the axiomatic approach,
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263
under the suitable conditions concerning two (fuzzy) set-valued operators, the uniqueness holds
in the following several generalized rough set models: crisp relation based rough set model,
fuzzy relation based rough set model, and (T, I) type fuzzy rough set model.
It is reasonable to expect that the main results presented in this paper are applied to both
theoretical and practical researches of generalized rough sets.
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Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China.
Email: [email protected]