On Some Topological Properties of Pessimistic Multigranular Rough Sets

Published Online July 2012 in MECS

The main observations of our traditional tools for formal modeling, reasoning and computing are crisp, deterministic and preciseness nature, which restrict their applicability in real life situations, led to the extension of the concept of crisp sets so as to model imprecise data and enhance their modeling power. One such approach to capture impreciseness was due to Pawlak [5, 6], who introduced the notion of Rough Sets, which is an excellent tool to capture impreciseness in data. The basic assumption of rough set theory is that human knowledge about a universe depends upon their capability to classify its objects. Classifications of a universe and equivalence relations defined on it are known to be interchangeable notions. So, for mathematical reasons equivalence relations were considered by Pawlak to define rough sets. A rough set is represented by a pair of crisp sets, called the lower approximation comprises of elements, which belong to it definitely and upper approximation comprises of elements, which are possibly in the set with respect to the available information.

To improve the modeling capability of basic rough sets several extensions have been made in different directions. One such extension is the rough sets based upon tolerance relations instead of equivalence relations. These rough sets are sometimes called incomplete rough set models. In the view of granular computing, classical rough set theory is researched by a single granulation. The basic rough set model has been extended to rough set model based on multigranulations (MGRS)  in [10],  where the set approximations are defined by using multi-equivalences on the universe. Using similar concepts, that is taking multiple tolerance relations instead of multiple equivalence relations; incomplete rough set model based on multi-granulations was introduced in [11]. Several fundamental properties of these types of rough sets have been studied [10, 11, 12].

Employing the notions of lower and upper approximations of rough sets, an interesting characterization of rough sets has been made by Pawlak in [6], where he introduced the types (originally called kinds) of rough sets. There are two different ways of characterizing rough sets; the accuracy coefficient and the topological characterization introduced through the notion of types. As mentioned by Pawlak himself [6], in practical applications of rough sets we combine both types of information about the borderline region, that is, of the accuracy of measure as well as the information about the topological classification of the set under consideration. Keeping this in mind, Tripathy and Mitra [16] have studied the types of rough sets by finding out the types of union and intersection of rough sets of different types. These results were extended to the context of optimistic multi granular rough sets by Tripathy et al [17]. In this paper, we study these results for the pessimistic multi granular context, which also remains the same for both the complete and incomplete cases.

• II.    Definitions And Notations

Let U be a universe of discourse and R be an equivalence relation over U. By U/R we denote the family of all equivalence classes of R, referred to as categories or concepts of R and the equivalence class of an element x e U is denoted by [x]R . By a knowledge base, we understand a relational system K = (U, P) , where U is as above and P relations over U. For any subset Q ( ^ ф ) c P, the intersection of all equivalence relations in Q is denoted by IND(Q) and is called the indiscernibility relation over Q. Given any X c U and R e IND (K), we associate two subsets, RX = U{Y e U / R : Y c X} and RX = U{Y e U / R: Y П X ^ ф}, called the R-lower and R-upper approximations of X respectively. The R-boundary of X is denoted by BNR (X) and is given by BN^ (X) = RX — RX. The elements of R X are those elements of U, which can certainly be classified as elements of X, and the elements of R X are those elements of U, which can possibly be classified as elements of X, employing knowledge of R. We say that X is rough with respect to R if and only if RX ^ RX, equivalently BNR (X) ^ ф. X is said to be R-definable if and only if  RX = RX ,  or

BN r ( X ) = ф .

In the view of granular computing (proposed by L. A. Zadeh), an equivalence relation on the universe can be regarded as a granulation, and a partition on the universe can be regarded as a granulation space [2, 3]. For an incomplete information system, similarly, a tolerance relation on the universe can be regard as a granulation, and a cover induced by the relation can be regarded as a granulation space. Several measures in knowledge base closely associated with granular computing, such as knowledge granulation, granulation measure, information entropy and rough entropy, were discussed in [2, 3, 4]. On research of rough set method based on multi-granulations, Y. H. Qian and J. Y. Liang brought forward a rough set model based on multigranulations [10], which is established by using multi equivalence relations. In [11] an extension of MGRS, rough set model based on multi tolerance relations in incomplete information systems is developed.

We define below the optimistic MGRS.

Definition 2.1: Let K= (U, R ) be a knowledge base, R be a family of equivalence relations, X c U and P,Q e R. We define the optimistic multi-granular lower approximation and upper approximation of X in U as

• (2.1)     P + Q X = U{ x / [ x ]_P c X or [ x ]₀ c X }

and

• (2.2)           P + QX = ( P + Q ( X C )) C

Another kind of multi-granular rough sets called pessimistic multi-granular rough sets was introduced by Quian et al [13]. Now, they call the above type of multi-granular rough sets as the optimistic multi-granular rough sets (which was introduced as the multi-granular rough sets [10]).

We define below the pessimistic multi-granular rough sets (PMGRS).

Definition 2.2: Let K= (U, R ) be a knowledge base, R be a family of equivalence relations, X c U and P,Q e R. We define the pessimistic multi-granular lower approximation and upper approximation of X in U as

• (2.3)       ( P + Q ) ^PX = U{ x /[ x ]p c X and [ x ]_g c X }

and

• (2.4)             ( P + Q ) ^PX = ( ( P + Q ) ^P ( X C )) C

We state below several properties of pessimistic multi-granular rough sets (PMGRS) from [13].

Property 2.1: Let K= (U, R ) be a knowledge base, R be a family of equivalence relations, X c U and P,Q e R. The following properties hold true.

P

• (2.4) ( P + Q ) ( X ) c X c ( P + Q ) ( X )

PP

• (2.5)    ( P + Q ) ( ф ) = ф = ( P + Q ) ( ф ), ( P + Q ) ( U ) = U = ( P + Q ) ( U )

• (2.6) ( P + Q ) P ( X C ) = (( P + Q ) P ( X )) C

• (2.7) ( P + Q ) P ( X ) = PXHQX

P

• (2.8) ( P + Q ) ( X ) = PXDQX

PP

• (2.9)    ( P + Q ) ( X ) = ( Q + P ) ( X ),( P + Q ) ( X ) = ( Q + P ) ( X)

Property 2.2: Let K= (U, R ) be a knowledge base, R be a family of equivalence relations, X,Y c U andP,Q e R. The following properties hold true.

• (2.10)    ( P + Q ) P ( X A Y ) = ( P + Q ) P X П ( P + Q ) P Y

P   PP

• (2.11)( P    + Q ) ( X U Y ) = ( P + Q ) X U( P + Q ) У

• (2.12)    ( P + Q ) P ( X U Y ) 5 ( P + Q ) P X U ( P + Q ) P Y

P   PP

• (2.13)(    P + Q ) ( X^Y ) c ( P + Q ) X П( P + Q ) У

• Next, we define PMGRS in incomplete information systems.

Definition 2.3: An information system is a pair

S = (U, A), where U is a non-empty finite set of objects, A is a non-empty finite set of attributes. For every a e A , there is a mapping a : U ^ V_a , where V is called the value set of a.

If V contains a null value for at least one attribute a e A , then S is called an incomplete information system. Otherwise, it is complete.

Definition 2.4:  let s = (U,A) be an incomplete information system, P c A an attribute set. We define a binary relation on U as follows

• (2.14)           SIM(P) = {(u,v) e U X U | V a e P,

a(u) = a(v) or a(u) = * or a(v) = *}.

In fact, SIM(P) is a tolerance relation on U, the concept of a tolerance relation has a wide variety of applications in classifications [1, 8].

It can be shown that SIM(P) = Q     SIM({A}).

Let S _p (u) denote the set {v e U | (u,v) e SIM(P)}.

S (u) is the maximal set of objects which are possibly indistinguishable by P with u.

Let U/SIM(P) denote the family sets {S _p (u)| u e U}, the classification or the knowledge induced by P. A member S (u) from U/SIM(P) will be called a tolerance class or an information granule. It should be noticed that the tolerance classes in U/SIM(P) do not constitute a partition of U in general. They constitute a cover of U, i.e., S (u) * ф for every u e U, and

S (u) = U.

u e U ^{p V 7}

Definition 2.5:  Let S = (U, A) be an incomplete information system, P,Q c A two attribute subsets, X c U, we define a lower approximation of x and a upper approximation of x in U by the following

• (2.15) ( P + Q ) P X = U {x | SIM _p (x) c X and

SIM (x) c X} and

• (2.15)              ( P + Q ) P (X) = ( ( P + Q ) P ( X c )) ^c

A Multi-granulation Rough Set can be classified into following four types:

Definition 2.6: Let K= (U, R ) be a knowledge base, R be a family of equivalence relations, X c U and P,Q e R. Then

• (2.16)    If ( P + Q ) P (X) * ф and ( P + Q ) P * U, then we say that X is pessimistic roughly P+Q-definable.

• (2.17)    If ( P + Q ) P (X) = ф and ( P + Q ) P * U, then we say that X is pessimistic internally P+Q-undefinable. (2.18). If ( P + Q ) P (X) * ф and ( P + Q ) P = U, then we say that X is pessimistic externally P+Q -undefinable.

• (2.19)    If ( P + Q ) P (X) = ф and ( P + Q ) P = U, then we say that X is pessimistic totally P+Q –undefinable.

• III.    Results

• 3.1    Table for type of X with respect to ( P + Q ) ^P

• 3.1.1    Example to prove entry (1,1)

In this section we shall find out the types of pessimistic multi granular rough sets (PMGRS). There are four sets of results accumulated in four tables. The first provides the type of a P+Q rough set from the types of its P and Q rough set types. The second table provides the types of the complement of a multi granular rough set. In the third table we obtain the types for the union of two multi granular rough sets of all possible types. Similarly we establish the types of the intersection of two multi granular rough sets of all possible types. These results will be useful for further studies in approximation of classifications and rule generation.

This subsection provides the type of a P+Q rough set from the types of its P and Q rough set types in the following table. Proofs with examples for some of the entries in the table are then given.

	Type of X with res			ect to Q
Type		T-1	T-2	T-3	T-4
of X	T-1	T-1	T-1	T-1	T-1
with	T-2	T-1	T-2	T-1	T-2
respect	T-3	T-1	T-1	T-3	T-3
to P	T-4	T-1	T-2	T-3	T-4

Let U = {a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈}.

U ¹ SIM ^{( P} ) = ^{ { a , a 7 }, ^{{ a} 2 , a 3 , a 4 , a 5 , a 6 , a 7 }, ^{a 8 ^{} }}

U I SIM ( Q ) = {{a_x , a₇ }{a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ?, a_% }, { a ₃, a ₄, a ₅, a ₆},{ a ₈}}

LetX = {a_x , a₃ , a₄ , a- }.

PX = {a_x , a ₇} * ф and PX = {a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇} * U .

Thus X is of Type - 1 w.r . t . P.

QX = {a_x , a ₇} * ф andQX = {a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇} * U .

Thus X is of Type - 1 w . r . t . Q .

( P + Q ) ^PX = {a_x , a ₇} * ф and

( P + Q ) ^P X = { a , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇} * U .

Thus X is of Type - 1 w . r . t . ( P + Q ) ^P .

• 3.1.2    Example to prove entry (1,3)

Let U = {a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈}.

U I SIM ( P ) = {{ a , a ₇ },{ a ₂, a ₃, a ₄, a ₅, a ₆, a ₇ },{ a ₈}}

UI SIM ( Q ) = {{ a , a ₇},{ a ₁, a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈},

{ a ₃, a ₄, a ₅, a ₆},{ a ₈}}

LetX = {a_x , a ₇, a ₈}.

PX = {a_x , a ₇, a ₈} * ф and PX = {a_x , a ₇, a ₈} * U .

Thus X is of Type - 1 w.r . t . P .

QX = {a_x , a ₇, a ₈} ^ ф and QX = U .

Thus X is of Type - 3 w . r . t . Q .

( P + Q ) ^P ( X ) = { a , a ₇, a ₈} ^ ф and

( P + Q ) ^PX = {a_x , a ₇, a ₈} ^ U .

Thus Xis of Type - 1 w . r . t . ( P + Q ) ^p

• 3.2    Table for type of X C with respect to ( P + Q ) ^P

• 3.2.1    Example to prove entry of row2

This subsection provides the types of complement of pessimistic multi granulation rough set in the following table. Proofs with examples for some of the entries in the table are then given.

X	XC
T-1	T-1
T-2	T-3
T-3	T-2
T-4	T-4

Let U = { a , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈}.

U I SIM ( P ) = {{a_x , a ₇},{ a ₂, a ₃, a ₄, a ₅, a ₆, a ₇},{ a ₈}}

Let X = { a ₂, a ₃, a ₄ }.

PX = ф and PX = { a ₂, a ₃, a ₄, a ₅, a ₆} ^ U .

Thus X is of Type - 2 w.r.t. P.

X     ^{{ a} 1 , ^a ₅ , a 6 , a 7 , a 8 ^} .

PXC = { ax, a7, a8} ^ ф and

PX^C = {a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈} = U .

Thus Xе is of Type - 3 w.r.t. P

• 3.3    Table for type of X U Y with respect to ( P + Q ) ^P

This subsection provides the types of union of two pessimistic multi granular rough sets with respect to P+Q in the following table. Proofs with examples for some of the entries in the table are then given.

Proof of entry (1, 1)

Suppose X and Y are both of Type-1. Then

(P + Q)PX ^ ф, (P + Q)PY * ф,(P + Q)PX * U and (P + Q)PY / U.From (2.12) it follows that

( P + Q ) P ( X U Y ) * ф .

But using (2.11) we see that

P

(P + Q) (X U Y)has both the possibilities of being equal or not equal to U.

So , X U Y can be of Type - 1 or of Type - 3.

и	P Type of Y with respect to ( P + Q )
Type of X with respect to ( P + Q ) P		T-1	T-2	T-3	T-4
	T-1	T-1/ T-3	T-1/T-3	T-3	T-3
	T-2	T-1/ T-3	T-1/T-2/ T-3/T-4	T-3	T-3/ T-4
	T-3	T-3	T-3	T-3	T-3
	T-4	T-3	T-3/T-4	T-3	T-3/ T-4

• 3.3.1    Examples to prove entry (1,1)

Let U = { a ,, a ,, a. , a. , a, , a, , a, , a„ }.

U I SIM ( P ) = {{ a_x , a ₇},{ a ₂, a ₃, a ₄, a ₅, a ₆, a ₇}, { a 8 }}

U I SIM ( Q ) = {{ a , a ₇},{ a , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈}, { a ₃, a ₄, a ₅, a ₆},{ a ₈}}

Case 1

LetX = { a ,, a ₇} and Y = { a₈ }. Then XUY = { a ,, a ₇, a ₈}.

( P + Q ) P X = { a , a ₇} * Ф and ( P + Q ) p X = { a i , a ₇} * U . Thus X is of Type — 1

( P + Q ) P Y = { a } * Ф and ( P+Q ) ^PY = { a } * U .

Thus Y is of Type — 1

( P + Q ) P ( XUY ) = { a , a ₇, a ₈} * ф and

( P + Q ) ^p ( XUY ) = { a_pa 7 , a 8 } * U .

Thus XUY is of Type - 1

Case 2

Let X = { a ,, a ₃, a ₄, a ₇} and Y = { a ₈, a ₄, a ₅}. Then XUY = { a_x , a ₃, a ₄, a ₅, a ₇, a ₈}.

( P + Q ) ^P X = { a_x , a ₇ } * ф and

( P + Q ) ^p X = { a_x , a ₇, a ₂, a ₃, a ₄, a ₅, a ₆ } * U .

Thus X is of Type — 1

( P + Q ) ^P Y = { a ₈ } * ф and

( P + Q ) ^p Y = { a ₈, a ₃, a 4 , a ₅, a 6 } * U .

Thus Y is of Type — 1

( P + Q ) P ( XUY ) = { a_x , a ₇, a ₈} * ф and

( P + Q ) ^p ( XUY ) = { a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈} = U .

Thus XUY is of Type — 3

Proof of entry (1, 3)

Let both X and Y be of Type 1 and Type 3. Then from the properties of type 1 and type 3

( P + Q ) P ( X ) * ф ,  ( P + Q ) P (Y) * ф ,

( P + Q ) ^P (X) * U and  ( P + Q ) ^P (Y) = U.

So, using (2.11) and (2.12) we get

( P + Q ) P (XUY) * ф and ( P + Q ) P (XUY) = U.

Hence XUY is of type 3 only.

The other cases can be similarly established.

• 3.3.2    Example to prove entry (1,3) in XUY table

Let U = { a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈}.

UI SIM ( P ) = {{ a 1 , a ₇},{ a 2 , a ₃, a ₄, a ₅, a 6 , a ₇},{ a 8 }}

U I SIM ( Q ) = {{ a , a ₇},{ a , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈}, { a ₃, a ₄, a ₅, a ₆},{ a ₈}}

LetX = { a_x , a ₃, a ₄, a ₇} andY = { a_x , a ₄, a ₅, a ₇, a ₈}.

( P + Q ) ^pX = { a , a ₇} * ф and ( P + Q ) ^pX * U .

Thus X is of Type — 1.

( P + Q ) ^PY = { a , a ₇, a ₈} * ф and ( P + Q ) ^pY = U .

Thus Y is of Type — 3.

XUY = { a_x , a ₃, a ₄, a ₅, a ₇, a ₈ }.

( P + Q ) ^p ( XUY ) = { a_x , a ₇, a ₈ } * ф and

( P + Q ) ^p ( XUY ) = U .

Thus XUY is of Type — 3.

• 3.3.3    Examples to prove entry (2,2) in XUY table

Let us take the following examples and provide proofs for first two cases.

Let U = { a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈}.

U I SIM ( P ) = {{ a_x , a ₇},{ a ₂, a ₃, a ₄, a ₅, a ₆, a ₇}, { a 8 }}

U I SIM ( Q ) = {{ a_x , a ₇},{ a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈}, { a ₃ , a ₄ , a ₅, a ₆},{ a ₈}}

Case 1

LetX = { a_x , a ₃, a ₄} and Y = { a ₇}.

( P + Q ) ^PX = ф and ( P + Q ) ^PX = { a , , a ₂, a 3 , a ₄, a ₅, a ₆, a 7 } * U .

Thus X is of Type — 2.

( P + Q ) ^pY = ф and ( P + Q ) ^pY = { a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇} * U .

Thus Y is of Type — 2.

XUY = { a_x , a ₃, a ₄, a ₇}.

( P + Q ) P ( XUY ) = { a_x , a ₇} * ф and

( P + Q ) ^p ( XUY ) = { a_x , a ₂, a ₃, a ₄, a ₅, a ₆} * U .

Thus XUY is of Type — 1.

Case 2

LetX = {a₃ , a₄ , a ₅, a₆ } and

Y = {a , a ₄, a ₅}.

( P + Q ) ^P X = ф and ( P + Q ) ^P X = { a , a ₃, a ₄, a ₅, a ₆} * U .

Thus Xis of Type — 2.

XUY = { a ₂, a ₃, a ₄, a ₆, a ₇}.

( P + Q ) ^P ( XUY ) = ф and ( P + Q ) ^p ( XUY ) = U .

Thus XUY is of Type — 4

• 3.4    Table for type of X П Y with respect to ( P + Q ) ^P

( P + Q ) ^PY = ф and ( P + Q ) ^pY = { a ₂, a ₃, a ₄, a ₅, a ₆}* U .

Thus Y is of Type — 2.

XUY = { a ₃, a ₄, a ₅, a ₆ }.

( P + Q ) ^P ( XUY ) = ф and

( P + Q ) ^p ( XUY ) = {a 2 , a 3 , a 4 , a ₅, a б } * U .

Thus XUY is of Type — 2

Then let us take the following examples and provide proofs for the next two cases.

Let U = { a_x , a ₂, a ₃, a₄a ₅, a ₆, a ₇, a ₈}

^U / SI^M ⁽ P )   ^{{{ a} i , ^a 7 ^}, ^{a 2 , a , ^a 4 , ^a 5 ^}, ^{a 6 , a ^}}

U I SIM ( Q ) = {{a_x , a ₇ },{ a ₁, a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈},

{ a ₄ , a ₅ , a ₆ },{ a ₆ , a ₈ }}

Case 3

Let X = { a ₃, a ₄, a ₇} and Y = { a ₁, a ₈}.

( P + Q ) ^P X = ф and ( P + Q ) ^P X = { a ₂, a ₃, a ₄, a ₅} * U .

Thus X is of Type — 2.

( P + Q ) ^P Y = ф and ( P + Q ) ^P Y = {a_x , a ₂, a ₃, a ₇, a ₈} * U . Thus Y is of Type — 2.

XUY = {a, , a ₃, a ₄, a ₆, a ₇}.( P + Q ) ^P ( XUY ) = {a , a ₇} * ф and ( P + Q ) ^p ( XUY ) = U .

Thus XUY is of Type — 3.

Case 4

Let X = { a ₃, a ₄, a ₇ } and Y = { a ₆, a ₂ }.

( P + Q ) ^P X = ф and ( P + Q ) ^P X ^ U .

Thus X is of Type — 2.

( P + Q ) ^PY = ф and ( P + Q ) ^PY ^ U .

Thus Y is of Type — 2.

This subsection provides the types of intersection of two pessimistic multi granular rough sets with respect to P+Q in the following table. Proofs with examples for some of the entries in the table are then given.

A	Type of Y with respect to ( P + Q )
Type of X with respect to ( p + Q ) P		T-1	T-2	T-3	T-4
	T-1	T-1/T-2	T-2	T-1/T-2	T-2
	T-2	T-2	T-2	T-2	T-2
	T-3	T-1/T-2	T-2	T-1/T-2/ T-3/T-4	T-2 /T-4
	T-4	T-2	T-2	T-2/T-4	T-2 /T-4

Proof of entry (1, 3)

Suppose X is of Type-1 and Y is of Type-3. Then

( P + Q ) ^P X * ф , ( P + Q ) ^P Y * ф ,

PP

( P + Q ) X * U and ( P + Q ) Y = U .

From (2.13) it follows that

( P + Q ) P ( X Q Y ) * U .

But using (2.10) we see that

( P + Q ) ^P ( X П Y ) has both the possibilities of being or not being equal to ф .

So X П Y can be of Type — 1 or Type — 2

• 3.4.1    Examples to prove entry (1,3)

Let U = {a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈}.

UI SIM ( P ) = {{ a 1, a ₇},{ a ₂, a ₃, a ₄, a ₅, a 6, a 7},

{ a 8}}

U I SIM ( Q ) = {{a_x , a₇}{a_x , a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈}, { a ₃, a ₄, a ₅, a ₆},{ a ₈}}

Case 1

LetX = { a ₂, a₃a ₄, a ₅, a ₆} andY = { a ₂, a₃a ₄, a ₅, a ₆, a ₇, a ₈}.

Then X n Y = { a ₂, a₃ a ₄, a ₅, a ₆}.

(P+Q)PX * ф and (P+Q)PX={a2, a3a4, a5, a6} * U.

Thus X is of Type - 1.

(P+Q)pY*ф and(P+Q)PY={a,,a2,a3a4,a5,a6,a7,a8}=U.

Thus Y is of Type - 3.

(P+Q)P (X n Y) * ф and (P+Q)P (X n Y) * U.

Thus X n Y is of Type - 1.

Case 2

LetX = { a₃a ₄, a ₅, a ₈} andY = { a , a ₂, a₃a ₄, a ₅, a ₆, a ₈}.

Then X n Y = { a₃a ₄, a ₅}.

( P + Q ) ^PX = { a ₈} * ф and ( P + Q ) ^PX = { a ₂, a₃a ₄, a ₅, a ₆, a ₈} * U .

Thus Xis of Type - 1.

( P + Q ) ^pY = { a ₈} * ф and ( P + Q ) P Y = { a , a ₂, a₃ a ₄, a ₅, a ₆, a ₇, a ₈}= U .

Thus Y is of Type - 3.

( P + Q ) ^P ( X n Y ) = ф and ( P + Q ) ^P ( X n Y ) * U .

Thus X n Y is of Type - 2.

Let X and Y be of Type 2 and Type 1 respectively. Then from the properties of type 2 and type 1 multi granular rough sets we get ( P + Q ) P (X) = ф ,

( P + Q ) ^P (Y) = ф , ( P + Q ) P (X) * U and ( P + Q ) ^P (Y) * U.

So using properties (2.10) and (2.13) we get ( P + Q ) ^P (X П Y) = ф and ( P + Q ) ^P (X П Y) * U. So, XY is of type 2. This completes the proof. The other cases can be established similarly.

• 3.4.2    Example to prove entry (2,1)

Let U = {ax, a2, a3, a4, a5, a6, a7, a8}.

U / SIM (P) = {{ax, a7 },{a2, a3, a4, a5, a6, a?}, {a8}}

UI SIM(Q) = {{a1,a7},{a1,a2,a3,a4,a5,a6,a7,a8}, {a3 ,a4,a5 ,a6},{a8}}

Proof of entry (2, 1)

LetX = { a₃a ₄, a ₅, a ₈} andY = { a₃a ₄, a ₅}.

Then X n Y = { a₃ a ₄, a ₅}.

(P + Q)P X = {a8} * ф and (P + Q)P X={a2, a3 a4, a5, a6, a8} * U.

Thus X is of Type -1.

( P + Q ) ^pY = ф and ( P + Q ) ^PY = { a ₂, a₃a ₄, a ₅, a ₆} * U .

Thus Y is of Type - 2.

( P + Q ) ^P ( X n Y ) = ф and ( P + Q ) ^P ( X n Y ) = { a ₂, a₃a ₄, a ₅, a ₆} * U .

Thus X n Y is of Type - 2

• IV.    Conclusion

Two types of multi-granular rough sets have been introduced in the literature ([10], [13]). Topological properties of optimistic multigranular rough sets were studied by Tripathy et al ([17]). In this paper we studied the topological properties of pessimistic multi granular rough sets with respect to the three set theoretic operations of union, intersection and complementation. The tables show that there are multiple answers to some of the cases as like as the case of basic rough sets. These multiple answers specify impreciseness and ambiguity in information that is available with the user to classify object of a universe. Thus the models discussed in this paper are rightly suitable for handling impreciseness in data in more effective and elegant manner using both types of multi-granular rough sets. Also, we provided examples in some cases to illustrate the fact that the multiple answers can actually occur. These results can be used in approximation of classifications and rule induction. Also our results hold true for both complete and incomplete pessimistic multigranulation systems