Composition of soft set relations and construction of transitive

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.7, 2012

Composition of Soft Set Relations and Construction of Transitive
Closure
A. M. Ibrahim*, M. K. Dauda and D. Singh
Department of Mathematics
Ahmadu Bello University
Zaria, Nigeria
Abstract
In [3] concepts of soft set relations, partition, composition and function are discussed. In this
paper, we present composition of relations in soft set context and give their matrix
representation. Finally, the concepts of reflexive, symmetric and transitive closure are
presented and show that construction of transitive closure in soft set satisfies Warshall’s
Algorithm.

1 Preliminaries and basic definitions
The origin of soft set theory could be traced to the work of Pawlak [6] in 1993 titled Hard and
Soft Set in Proceeding of the International EWorkshop on rough sets and knowledge
discovery at Banff. His notion of soft sets is a unified view of classical, rough and fuzzy sets.
This motivated D. Molodtsov’s work [2] in 1999 titled soft set theory: first result. Therein,
the basic notions of the theory of soft sets and some of its possible applications were
presented. For positive motivation, the work discusses some problems of the future with
regards to the theory.
Let be a universal set and let E be a set of parameters (each parameter could be a word or a
sentence). Let denotes the power set of . In [2] and [5], a pair is called a soft set
over a given universal set , if and only if is a mapping of a set of parameters , into the
power set of . That is, . Clearly, a soft set over is a parameterized family
of subsets of a given universe . Also, for any , is considered as the set of
approximate element of the soft set .
Example 1
Let be the set of Cars under
consideration, be a set of parameters.

{ expensive, beautiful, manual gear, cheap, automatic gear,
in good repair, in bad repair}.

The soft set describes the attractiveness of the cars. For more examples, see [1], [2],
[3], [4] and [5].
* Corresponding author: adekubash1@gmail.com

Definition 1: A soft set over a universe is said to be null soft set denoted by , if
, .
Definition 2: A soft set over a universe is called absolute soft set denoted by ,
if , .

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Vol.2, No.7, 2012

Definition 3: Let = { , , } be a set of parameters. The not-set of E denoted
by is defined as ={ , , , ).
Definition 4: The complement of a soft set denoted by is defined as
.
Where: is a mapping given by
is called the soft complement function of .
Clearly, (i) and (ii)
Definition 5: For any two soft sets and over a common universe , we say that
is a soft subset of if
(i) , and
(ii) , and are identical approximations.

We write .
is said to be a soft super set of , if is a subset of and it is denoted by
.
Definition 6: Two soft sets and over a common universe are said to be soft
equal if is a soft subset of and is a soft subset of . It is denoted by

Definition 7: If and are two soft sets then denoted by
is defined as , where ,
.
Definition 8: If and are two soft sets then denoted by
is defined by
Where,

2 Soft set relations
Let and be two soft sets over , then the Cartesian product of and
is define as where and
where i.e. ; where and
[3].
The Cartesian product of three or more nonempty soft sets can be defined by generalizing the
definition of the Cartesian product of two soft sets. The Cartesian
of the nonempty soft sets is the soft
sets of all ordered n-tuples where

Example 2

Let soft sets and describe the “cost of the houses” and “attractiveness of houses”
respectively.

Suppose that

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Vol.2, No.7, 2012

.

Let , , , and respectively stand for very costly, costly, cheap, beautiful and in the
green surrounding.

Suppose and defined as following:

Now, where a typical element will look like

.

Relation in soft set is defined in terms of ordered pairs.

Definition 9: Let and be two soft sets over , then a relation from to
is a soft subset of A relation from to is of the form
) where and Any subset of is
called a relation on In an equivalent way, we can define the relation on the soft set
in the parameterized form as follows.

If then if and only if

Definition 10: Let be a soft set relation from to then the domain of denoted
as , is defined as the soft set where

for some and

The range of denoted by , is defined as the soft set , where and
for some and where .

Definition 11: Let be any soft set. Then and are soft subsets of
and hence are relation on called universal relation and empty relation
respectively. Thus for any relation on we have

Definition 12: The identity relation on any soft set is defined as follows
iff

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Vol.2, No.7, 2012

3 Composition of soft set relation
Definition 13: Let and be three soft sets. Let be a soft set relation
from to and be another soft set relation from to , then the
composition of and is a new soft set relation from to expressed as and
is defined as follows;

If is in and is in then iff there is some in
such that and .

Example 3

Let and Let and be
soft set relation defined respectively from to and to as

and

Then . In general

3.2 Types of soft set relation
Definition 14: Let be a relation on , then

(i) is reflexive if
(ii) is symmetric if

(iii) is anti-symmetric if whenever
and then
(iv) is transitive if

(v) is an equivalence relation if it is reflexive,
symmetry and transitive.
Definition 15: The inverse of a soft set relation denoted by is defined by

Theorem Let be soft set relation from to and be a soft set relation from
to . Then .

Proof: See [3]

4. Composition of soft set relation using matrices

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Vol.2, No.7, 2012

In this section, we give matrix representation of composition of soft set relations. If the
resultant matrices from the relations are compatible, the representation is straightforward
otherwise an adjustment is made to the matrices before the composition.
Example 3

Let and Let and
be soft set relation defined respectively from to and to as

and

By definition 13,

The matrices representation of , and respectively are,

, and

Multiplying the two matrices, and we obtain the matrix

The non-zero entries in these matrices indicate the elements that are related. Thus,
and have the same nonzero entries.

Hence,

Example 4

Let and Let and
be soft set relation defined respectively from to and to as

and

By definition 13, …{1}

Matrix representation of and respectively are,

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Vol.2, No.7, 2012

and

Clearly, is of order and is of order which are not compatible to
multiplication. Therefore, an adjustment is made to make the two compatible by creating a
third column fill with zeros in .

Let and

is now of order while the order of remain unchanged, thus compatible to matrix
multiplication. Therefore,

Let

From the matrices result, we have

…{2}

Since {1} and {2} yield the same result, hence the adjustment is valid.
Remark 1: Composition of soft set relations is not commutative in general just like matrix
multiplication.

5 Closure of a relation
Suppose that is a relation on a soft set may or may not have some property , such
as reflexivity, symmetry, or transitivity. If there is a relation with property containing
such that is a sub soft set of every relation with property containing , then is called the
closure of with respect to . The closure of a relation with respect to a property may or may
not exist.

5.1 Construction of closures of a relation
Reflexive Closure
Reflexive closure of can be formed by adding to all pairs of the form with
not already in The addition of these pairs produces a new relation that is
reflexive, contains , and is contained within any reflexive relation containing

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Vol.2, No.7, 2012

Definition 17: The reflexive closure of equals where
is the diagonal relation on

Example 5

The relation on the
soft set with is not reflexive. To make this reflexive relation
containing R is simply done by adding and to , since
these are the only pairs of the form that are not in Clearly, this new relation
contains R. Furthermore, any reflexive relation that contains must also contains
and Because this relation contains , is reflexive, and is
contained with every reflexive relation that contain , it is called the reflexive closure of .

Symmetry Closure
The symmetry closure of a relation is constructed by adding all ordered pairs of the form
, where is in the relation, that are not already present in
Adding these pairs produces a relation that is symmetric relation that contains

Definition 18: The symmetric closure of a relation is obtained by taking the union of relation
with its inverse i.e. where

Example 6
Let R be a relation such that

on the soft set with is not symmetric. To make symmetric, we need
to add and since these are the only pair of the form
with that are not in These new relation is symmetric and
contains Furthermore, any symmetric relation that contains must contain this new
relations, since a symmetric relations that contains must contain and
Consequently, this new relation is the symmetric closure of

Transitive closure
The construction of transitive closure of a relation is complicated than that of reflexive or
symmetric closure. The transitive closure of a relation can be found by adding new ordered
pairs that must be present and then repeating this process until no new ordered pairs are
needed.

is said to be transitive closure of if it satisfies the following conditions (i) is transitive
(ii) (iii) is the smallest transitive relation containing

Definition 19: Let be a relation on a soft set We define

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Vol.2, No.7, 2012

Properties of closures
Let be a relation on a soft set with elements. Then

(i) transitive ,

(ii) , where is the zero-one matrix of the relation
(iii) and , where and are relations on
with zero-one matrices and .

Example 7

Suppose is a relation on with , where
the zero-one matrix for is given by

Thus since

Now

Reading from the zero-one matrix, we see that

is the

Warshall’s Algorithm
Warshall’s algorithm is an efficient method for computing the transitive closure of a relation.
The following illustrate the algorithm for computing the transitive closure of a relation.

Step I: let be a given relation

Calculate

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If then stop transitive

Step II: let

Calculate

If stop is the transitive closure of

…
In general,

Calculate

If stop, is the transitive closure of

Continue otherwise.
This algorithms holds for constructing transitive closure of a soft set relation using zero-one
matrix. For example, we use this algorithm on example 7, where
.

The zero-one matrix for is

, hence we calculate

Therefore, is the transitive closure of which is the same as in example 7.

6 Conclusions
Soft set has potential applications in several directions, such as smoothness of functions,
game theory, operations research, Riemann-integral, measurement theory[1] and [2].
Transitive closure is applied in modeling, networking and in operations research hence
developing transitive closure in soft set context is a fruitful exercise.

Reference

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Vol.2, No.7, 2012

[1] A Sezgin A. O. Atag in , On operation of soft sets. Computers and Mathematics with
Applications. Volume 57, 2009, Page 1547-1553.
[2] D. Molodtsov. Soft Set Theory - First Results. Computers and Mathematics with
Applications Volume 37, 1999, 19-31

[3] K. V. Babitha and J. J. Sunil. Soft Set Relations and Functions. Computers and
Mathematics with Applications, 60 (2010) 1840-1849

[4] M. Irfan Ali et el. On some new operation in soft sets theory. Computers and
Mathematics with Applications.Volume 57, 2009, Page 1547- 1553.

[5] P.K. Maji, R. Biswas and A. R. Roy. Soft Set Theory. Computers and Mathematics
with Applications. Volume 45, 2003, Page 555-562

[6] Z. Pawlak, Hard and Soft Sets, Proceeding of The International EWorkshop on Rough
Sets and Knowledge Discovery, Baniff, 1993.

107

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