The notion of fuzzy sets was first introduced by Zadeh [1]. After that, many studies attempted to generalize the fuzzy set by using various approaches. Pawlak [2] introduced the concept of rough sets, Nanda and Majumda [3] and Coker [4] proposed the idea of fuzzy rough sets. Atanassov [5] introduced the idea of intuitionistic fuzzy sets. All these concepts provide useful means of expressing vagueness in real environments.

Combining the concepts of fuzzy rough sets and intuitionistic fuzzy sets, Samanta and Mondal [6] proposed the idea of intuitionistic fuzzy rough sets. By introducing further generalization, the authors [7, 8] also conducted a study on the intuitionistic fuzzy bitopology and intuitionistic smooth bitopology. Moreover, the categorical properties of the intuitionsitc fuzzy topological spaces were studied by the same research group [9–11].

Many attempts at combining fuzziness and roughness have been made. In [12], the measure of fuzziness in rough sets is provided and studied. In [13] a general framework for the study of fuzzy rough sets is presented, in which both constructive and axiomatic approaches are made. Lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined by Zhou et al. [14, 15]. Several important properties of intuitionistic fuzzy approximation operators are examined by many researchers including us [16–19].

However, the properties of the intuitionistic fuzzy rough sets are very complicated and inadequate in the sense of the extension of intuitionistic properties. This is because of the unnaturalness of the definition of fuzzy rough sets. For example, the double complement of a fuzzy rough set is different from itself. The property that the double complement of a set becomes the set itself is one of the essential properties of Boolean algebra. Hence this flaw is critical in expanding the related theory. In order to overcome this unnaturalness, we need a new approach to intuitionistic fuzzy rough sets.

In this paper, we introduce a new definition of intuitionistic fuzzy rough sets and investigate important properties about the image and inverse image of an intuitionistic rough sets under a mapping. This new approach enables us to manipulate fuzzy rough sets more simply and easily. All the results obtained from this new definition are different from the results in other papers, and will be proven useful in expanding the related theory.

In [3], the definition of fuzzy rough sets has been introduced. The paper said:

“we shall consider ( , ) to be a rough universe where is a nonempty set and is a Boolean subalgebra of the Boolean algebra of all subsets of . Also consider a rough set with X_L ⊆ X_U. A fuzzy rough set in X is an object of the form

A=(AL,AU),

where A_L and A_U are characterized by a pair of maps A_L : X_L → L and A_U : X_U → L with A_L(x) ≤ A_U(x), for all x ∈ X_L, where (L,≤) is a fuzzy lattice.”

Furthermore, the complement Ā of a fuzzy rough set A = (A_L, A_U) is defined by (Ā)_L(x) = (A_U>L)′(x), ∀x ∈ X_L and (Ā)_U(x) = (A_L<U)′(x), ∀x ∈ X_U,

where A_U>L(x) = A_U(x), ∀x ∈ X_L and

AL<U(x)={AL(x),if x∈XL,∨{AL(x)∣x∈XL},if x∈XU-XL.

Unfortunately, the double complement of a fuzzy rough set A is different from A, because X_L and X_U are different, i.e., X_L ≤ X_U, The property that double complement of a set becomes the set itself is one of the essential properties of Boolean algebra. Hence this flaw is critical in expanding the related theory. Thus we are going to introduce the new definition of a fuzzy rough set by weakening the condition of the old definition. Then the properties we obtain in this paper are different from the results in the above paper.

Definition 2.1

Ler X be an underlying set and (L, ≤) a fuzzy lattice. A fuzzy rough set in X is an object of the form

A=(AL,AU),

where A_L and A_U are defined by a pair of maps A_L : X → L and A_U : X → L with A_L(x) ≤ A_U(x), for all x ∈ X. Furthermore 0 = (0_L, 0_U) the null fuzzy rough set and 1 = (1_L, 1_U) the whole fuzzy rough set in X,

Definition 2.2

For any two fuzzy rough sets A = (A_L, A_U) and B = (B_L, B_U) of X, we define the following:

• A ⊆ B iff A_L(x) ≤ B_L(x) and A_U(x) ≤ B_U(x) for all x ∈ X.

• A = B iff A ⊆ B and B ⊆ A.

If {A_i | i ∈ J} is a family of fuzzy rough sets in X, with A_i = ((A_i)_L, (A_i)_U), then we define the following:

• (∪_i A_i)_L(x) = ∨ (A_i)_L(x) and (∪_i A_i)_U(x) = ∨ (A_i)_U(x) for all x ∈ X.

• (∩_i A_i)_L(x) = ∧ (A_i)_L(x) and (∩_i A_i)_U(x) = ∧ (A_i)_U(x) for all x ∈ X.

Definition 2.3

The complement Ā = ((Ā)_L, (Ā)_U) of a fuzzy rough set A = (A_L, A_U) in X is defined by (Ā)_L(x) = (A_U(x))′ and (Ā)_U(x) = (A_L(x))′ for all x ∈ X.

For any fuzzy rough set A = (A_L, A_U) and for any x ∈ X,

(A¯)L(x)=(AU(x))′≤(AL(x))′=(A¯)U(x).

So, Ā is again a fuzzy rough set. Furthermore, for any fuzzy rough set A = (A_L, A_U), we have the double complement property, i.e., (A¯)¯=A.

Theorem 2.4

Let A be any fuzzy rough set in X, then we have the following properties:

• 0 ⊆ A ⊆ 1.

• (0_)¯=1,(1_)¯=0_.

Theorem 2.5

If A, B, C, D and B_i, i ∈ J are fuzzy rough sets in X, then we have the following properties:

• A ⊆ B and C ⊆ D implies

  A ∪ C ⊆ B ∪ D and A ∩ C ⊆ B ∩ D.

• A ⊆ B and B ⊆ C implies A ⊆ C.

• A ∩ B ⊆ A, B ⊆ A ∪ B.

• A ∪ (∩_i B_i) = ∩_i(A ∪ B_i), A ∩ (∪_i B_i) = ∪_i(A ∩ B_i).

• A ⊆ B implies B̄ ⊆ Ā.

• ∩iBi¯=∪iBi¯,∪iBi¯=∩iBi¯

Definition 2.6

Let f : X → Y be a mapping and A = (A_L, A_U) a fuzzy rough set in X. The image f(A) = ((f(A))_L, (f(A))_U) of A under f is defined by, for any y ∈ Y

(f(A))L(y) =f(AL)(y) ={0_L,if f-1(y)=∅,∨{AL(x)∣x∈f-1(y)},otherwise,(f(A))U(y) =f(AU)(y) ={0_U,if f-1(y)=∅,∨{AU(x)∣x∈f-1(y)},otherwise.

Let B = (B_L, B_U) be a fuzzy rough set in Y . Then the inverse image f⁻¹(B) = ((f⁻¹(B))_L, (f⁻¹(B))_U) of B under f is defined by, for any y ∈ Y

(f-1(B))L(x)=f-1(BL)(x)=BL(f(x)),(f-1(B))U(x)=f-1(BU)(x)=BU(f(x)).

The image of A under f is well-defined, because

f(AL)(y)=∨{AL(x)∣x∈f-1(y)}≤∨{AU(x)∣x∈f-1(y)}=f(AU)(y).

Moreover, the inverse image of B under f is well-defined, because

f-1(BL)(x)=BL(f(x))≤BU(f(x))=f-1(BU)(x).

Theorem 2.7

If f : X → Y is a mapping, then for any fuzzy rough sets A and B in X, we have the following:

• A ⊆ B implies f(A) ⊆ f(B).

• If f is surjective, then f(A¯)⊇f(A)¯.

• f(∪_i∈J A_i) = ∪_i∈J f(A_i).

• f(∩_i∈J A_i) ⊆ ∩_i∈J f(A_i).

Example 2.8

Let X = {1, 2, 3}, Y = {a, b, c}. Let f : X → Y be a mapping with f(1) = a, f(2) = f(3) = c. Then f is not surjective. Consider a fuzzy rough set

A=({1/0.4,2/0.3},{1/0.4,2/0.4}).

Then

A¯=({1/0.6,2/0.6},{1/0.6,2/0.7}).

So

f(A¯)=({a/0.6,b/0,c/0.6},{a/0.6,b/0,c/0.7}).

But

f(A)=({a/0.4,b/0,c/0.3},{a/0.4,b/0,c/0.4}),

and so

f(A)¯=({a/0.6,b/1,c/0.6},{a/0.6,b/1,c/0.7}).

Hence f(A¯)⊆f(A)¯.

Remark 2.9

If f : X → Y and g : Y → Z are mappings, then for any fuzzy rough set C in Z, the inverse image of C under g ○ f is defined by (g ○ f)⁻¹(C) = f⁻¹(g⁻¹(C)) where g ○ f is the composition of g and f.

Theorem 2.10

If f : X → Y is a mapping, then for all fuzzy rough sets B, C, B_i, i ∈ J in Y , we have the following:

• f-1(B¯)=f-1(B)¯.

• B ⊆ C implies f⁻¹(B) ⊆ f⁻¹(C).

• f⁻¹(∩_i B_i) = ∩_i f⁻¹(B_i).

• f⁻¹(∪_i B_i) = ∪_i f⁻¹(B_i).

Theorem 2.11

If f : X → Y is a mapping, then for any fuzzy rough set A in X and B in Y , we have the following:

• B ⊇ f(f⁻¹(B)).

• A ⊆ f⁻¹(f(A)).

Definition 3.1

If A = (A_L, A_U) and B = (B_L, B_U) are two fuzzy rough sets in X with B ⊆ Ā, then the ordered pair (A, B) is called an intuitionistic fuzzy rough set (briefly IF rough set) in X. The condition B ⊆ Ā is called the intuitionistic condition (briefly IC).

Remark 3.2

In [6], the IC is that B ⊆ Ā and A ⊆ B̄. But by the definition of IF rough set in this paper, we get the followings. If B ⊆ Ā, then B_L(x) ≤ (Ā)_L(x) = (A_U(x))′. i.e. B_L(x) ≤ (A_U(x))′ and B_U(x) ≤ (A_L(x))′. So A_U(x) ≤ (B_L(x))′ and A_L(x) ≤ (B_U(x))′. Thus A ⊆ B̄. Moreover, the converse is clear. Therefore B ⊆ Ā iff A ⊆ B̄.

Definition 3.3

0* = (0, 1) and 1* = (1, 0) are called the null IF rough set and the whole IF rough set in X, respectively. Clearly 0*¯=1* and 1*¯=0*.

We denote by IFRS(X) the collection of all IF rough sets in X. Usually we shall use letters X, Y, Z, … to denote sets, letters A, B, C, … to denote fuzzy rough sets, and letters P, Q, R, … to denote IF rough sets.

Definition 3.4

53.4 Let P = (A, B) = ((A_L, A_U), (B_L, B_U)) and Q = (C, D) = ((C_L, C_U), (D_L, D_U)) be two IF rough sets in X. We define the following:

• P ⊆ Q iff A ⊆ C and B ⊇ D.

• P = Q iff P ⊆ Q and P ⊇ Q.

• The complement P̄ of P = (A, B) in X, is defined by P̄ = (B, A).

• For IF rough sets P_i = (A_i, B_i) in X, i ∈ J, we define

∪i∈JPi=(∪i∈JAi,∩i∈JBi),∩i∈JPi=(∩i∈JAi,∪i∈JBi).

Remark 3.5

Let P_i = (A_i, B_i) be an IF rough set for all i ∈ Δ, then Ai⊆Bi¯ for all i ∈ Δ. For any k ∈ Δ, we have

Ak⊆Bk¯⊆∪i∈ΔBi¯=∩i∈ΔBi¯.

Thus ∪i∈ΔAi⊆∩i∈ΔBi¯. Hence ∪_i∈ΔP_i = (∪_i∈ΔA_i, ∩_i∈ΔB_i) is also an IF rough set.

Similarly, for any k ∈ Δ, ∩iAi⊆Ak⊆Bk¯. So ∩iAi⊆∩iBi¯=∪iBi¯. Hence ∩_i P_i = (∩_i A_i, ∪_i B_i) is also an IF rough set.

Theorem 3.6

Let P = (A, B), Q = (C, D), R = (E, F) and P_i = (A_i, B_i), i ∈ J be IF rough sets in X, then we have the following:

• P ∩ P = P = P ∪ P.

• P ∩ Q = Q ∩ P, P ∪ Q = Q ∪ P

• (P ∩ Q) ∩ R = P ∩(Q ∩ R), (P ∪ Q) ∪ R = P ∪(Q ∪ R).

• P ∩ Q ⊆ P, Q ⊆ P ∪ Q.

• P ⊆ Q and Q ⊆ R implies P ⊆ R.

• P_i ⊆ Q, ∀i ∈ J implies ∪_i P_i ⊆ Q.

• Q ⊆ P_i, ∀i ∈ J implies Q ⊆ ∩_i P_i.

• Q ∪ (∩_i P_i) = ∩_i(Q ∪ P_i).

• Q ∩ (∪_i P_i) = ∪_i(Q ∩ P_i).

• (P¯)¯=P.

• P ⊆ Q iff Q̄ ⊆ P̄.

• ∪iPi¯=∩iPi¯,∩iPi¯=∪iPi¯.

Definition 3.7

Let f : X → Y be a mapping. Let P = (A, B) = ((A_L, A_U), (B_L, B_U)) be an IF rough set in X. Then we define an image of P under f by

f(P)=(f(A),f(B¯)¯),

where f(A) = (f(A_L), f(A_U)) and f(B¯)¯=((f(B¯)¯)L,(f(B¯)¯)U) is defined by (f(B¯)¯)L=f((B¯)L)¯ and (f(B¯)¯)U=f((B¯)U)¯.

We already know that f(A)(y) = (f(A_L)(y), f(A_U)(y)) = (∨{A_L(x) | x ∈ f⁻¹(y)}, ∨{A_U(x) | x ∈ f⁻¹(y)}) for any fuzzy rough set A in X and a function f : X → Y .

Theorem 3.8

By the above definition, we have the following conclusion:

For any fuzzy rough set B in X and a function f : X → Y , We have f(B¯)¯(y)=(∧{(BL(x))∣x∈f-1(y)},∧{(BU(x))∣x∈f-1(y)}), ∧{(B_U(x) | x ∈ f⁻¹(y)}).

Remark 3.9

We will prove that the image of any IF rough set P = (A, B) satisfies the IC.

(f(A))L(y)=f(AL)(y)=∨{AL(x)∣x∈f-1(y)}≤∨{(B¯)L(x)∣x∈f-1(y)}=∨{(BU(x))′∣x∈f-1(y)}=(∧{(BU(x))∣x∈f-1(y)})′=((f(B¯)¯)U(y))′=((f(B¯)¯)¯)L(y).

The proof of the upper part is similar. So, if P = (A, B) be an IF rough set in X, then f(P)=(f(A),f(B¯)¯) is also an IF rough set in Y .

Remark 3.10

By the above theorem, we have the following conclusion:

For any IF rough set P = (A, B) in X and any map f : X → Y , The image f(P) = (f(A), f(B)) of P under f is actually expressed as follows:

For any y ∈ Y

f(P)(y)=(∨{A(x)∣x∈f-1(y)},∧{B(x)∣x∈f-1(y)}),

where

∨{A(x)∣x∈f-1(y)}={0_,if         f-1(y)=∅,(∨{AL(x)∣x∈f-1(y)},∨{AU(x)∣x∈f-1(y)}),if         f-1(y)≠∅,∧{B(x)∣x∈f-1(y)}={1_,if         f-1(y)=∅,(∧{BL(x)∣x∈f-1(y)},∧{BU(x)∣x∈f-1(y)}),if         f-1(y)≠∅.

Definition 3.11

Let f : X → Y be a mapping and Q = (C, D) = ((C_L, C_U), (D_L, D_U)) an IF rough set in Y . Then we define an inverse image of Q under f by

f-1(Q)=(f-1(C),f-1(D)),

where f⁻¹(C) = ((f⁻¹(C))_L, (f⁻¹(C))_U) = (f⁻¹(C_L), f⁻¹(C_U)) and f⁻¹(D) = ((f⁻¹(D))_L, (f⁻¹(D))_U) = (f⁻¹(D_L), f⁻¹(D_U)).

Remark 3.12

f-1(CL)(x)=CL(f(x))≤(D¯)L(f(x))=f-1((D¯)L)(x)=(f-1(D)¯)L(x). The upper part is also hold. Hence, the inverse image of an IF rough set under f is also an IF rough set.

Theorem 3.13

Let f : X → Y be a mapping. For any IF rough set P and Q on X, we have the following:

• P ⊆ Q implies f(P) ⊆ f(Q).

• If f is surjective, then f(P¯)⊇f(P)¯.

Definition 3.14

If f : X → Y and g : Y → Z are mappings, then the inverse image of IF rough set W in Z under (g ○ f) is defined by (g ○ f) ⁻¹(W) = f⁻¹(g⁻¹(W)) where g ○ f is the composition of g and f.

Theorem 3.15

Let f : X → Y be a mapping. For any IF rough set R, S and R_i, i ∈ J in Y , we have the following:

• f-1(R¯)=f-1(R)¯.

• R ⊆ S implies f⁻¹(R) ⊆ f⁻¹(S).

• f⁻¹(∪_i R_i) = ∪_i f⁻¹(R_i).

• f⁻¹(∩_i R_i) = ∩_i f⁻¹(R_i).

Theorem 3.16

Let f : X → Y be a mapping. For any IF rough set P in X and R in Y , we have the following:

• R ⊇ f(f⁻¹(R)).

• P ⊆ f⁻¹(f(P)).

Example 3.17

Let X = {1, 2, 3}, Y = {a, b, c}. And let f : X → Y be a mapping with f(1) = a, f(2) = f(3) = c. Consider an IF rough set R = (C, D) = (({a/0.4, b/0.3}, {a/0.4, b/0.4}), ({a/0.2, b/0.4}, {a/0.3, b/0.4})). Then we have R(b) = ((0.3, 0.4), (0.4, 0.4)). But f(f⁻¹(R))(b) = ((0, 1), (1, 0)). Thus R ≠ f(f⁻¹(R)).

Example 3.18

Let X = {1, 2, 3}, Y = {a, b, c}. And let f : X → Y be a mapping with f(1) = a, f(2) = f(3) = c. Consider an IF rough set P = (A, B) = (({1/0.4, 2/0.3}, {1/0.4, 2/0.4}), ({1/0.2, 2/0.4}, {1/0.3, 2/0.4})). Then we have P(3) = ((0, 1), (1, 0)). But f⁻¹(f(P))(3) = ((0.3, 0.4), (0.4, 0.4)). Thus P ≠ f⁻¹(f(P)).

Theorem 3.19

Let f : X → Y be a mapping. For any IF rough sets P, Q in X, we have f(P ∪ Q) = f(P) ∪ f(Q).

Corollary 3.20

If P₁, P₂, · · ·, P_n are IF rough sets in X, then

f(P1∪P2∪⋯∪Pn)=f(P1)∪f(P2)∪⋯∪f(Pn).

Theorem 3.21

Let f : X → Y be a mapping. For any IF rough sets P, Q in X, we have f(P ∩ Q) ⊆ f(P) ∩ f(Q).

Corollary 3.22

If f : X → Y is one-one, then clearly f(P ∩ Q) = f(P) ∩ f(Q).

But in general f(P ∩ Q) ≠ f(P) ∩ f(Q), which can be shown by the following example.

Example 3.23

Let X = {x, y} and Y = {a} and a mapping f : X → Y be defined by f(x) = f(y) = a. Let P = (({x/0.4, y/0.3}, {x/0.4, y/0.4}), ({x/0.2, y/0.4}, {x/0.3, y/0.4})), and Q=(({x/0.3, y/0.4}, {x/0.3, y/0.4}), ({x/0.3, y/0.2}, {x/0.3, y/0.3})). Clearly P, Q are IF rough sets in X. And P ∩ Q = (({x/0.3, y/0.3}, {x/0.3, y/0.4}), ({x/0.3, y/0.4}, {x/0.3, y/0.4})).

Thus

f(P∩Q)=(({a/0.3},{a/0.4}),({a/0.3},{a/0.3})),f(P)=(({a/0.4},{a/0.4}),({a/0.2},{a/0.3})),f(Q)=(({a/0.4},{a/0.4}),({a/0.2},{a/0.3})).

So f(P)∩f(Q) = (({a/0.4}, {a/0.4}), ({a/0.2}, {a/0.3})). Hence f(P ∩ Q) ≠ f(P) ∩ f(Q).

Theorem 3.24

Let f : X → Y be a mapping. For any IF rough set P_i, i ∈ J, we have the following:

• f(∪_i∈J P_i) = ∪_i∈J (P_i).

• f(∩_i∈J P_i) ⊆ ∩_i∈J (P_i).