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Studies in Fuzziness and Soft Computing
400th Volume of STUDFUZZ · 400th Volume of STUDFUZZ · 400th Volume of STUDFUZZ · 400th Volume of STUDFUZZ
Sunil Jacob John
Soft Sets
Theory and Applications
Volume 400
Series Editor
Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences,
Warsaw, Poland
The series “Studies in Fuzziness and Soft Computing” contains publications on
various topics in the area of soft computing, which include fuzzy sets, rough sets,
neural networks, evolutionary computation, probabilistic and evidential reasoning,
multi-valued logic, and related fields. The publications within “Studies in Fuzziness
and Soft Computing” are primarily monographs and edited volumes. They cover
significant recent developments in the field, both of a foundational and applicable
character. An important feature of the series is its short publication time and
world-wide distribution. This permits a rapid and broad dissemination of research
results.
Indexed by ISI, DBLP and Ulrichs, SCOPUS, Zentralblatt Math, GeoRef, Current
Mathematical Publications, IngentaConnect, MetaPress and Springerlink. The books
of the series are submitted for indexing to Web of Science.
More information about this series at http://www.springer.com/series/2941

Sunil Jacob John
Soft Sets
Theory and Applications
123
Sunil Jacob John
Department of Mathematics
National Institute of Technology Calicut
Calicut, Kerala, India
ISSN 1434-9922 ISSN 1860-0808 (electronic)

ISBN 978-3-030-57653-0 ISBN 978-3-030-57654-7 (eBook)
https://doi.org/10.1007/978-3-030-57654-7
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
Switzerland AG 2021
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transmission or information storage and retrieval, electronic adaptation, computer software, or by similar
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The use of general descriptive names, registered names, trademarks, service marks, etc. in this
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the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
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The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Dedicated to
My beloved father Late Mr. Jacob John
&
My respected Guru Prof. T. Thrivikraman.
Foreword
G. J. Klir has stated that among the various paradigmatic changes in science and
mathematics in the twentieth century, one such change concerned the concept of
uncertainty. In science, this change has been manifested by a gradual transition
from the traditional view, which states that uncertainty is undesirable in science and
should be avoided by all possible means, to an alternative which is tolerant of
uncertainty and insists that science cannot avoid it. Uncertainty is essential to
science and has great utility. An important point in the evolution of the modern
concept of uncertainty was a publication of a seminal paper by Lotfi Zadeh.
Soft set theory was proposed by Molodtsov in 1999 to deal with uncertainty in a
parametric manner. A soft set is a parameterized family of sets, intuitively soft
because the boundary of the set depends on the parameters. One notion of a set is
the concept of vagueness. This vagueness or the representation of imperfect
knowledge has been a problem for a long time for philosophers, logicians, and
mathematicians. Recently, it became a crucial issue for computer scientists par-
ticularly in the area of artificial intelligence. To handle situations like this, many
tools have been suggested. They include fuzzy sets, multisets, rough sets, soft sets,
and many more.
Molodtsov proposed soft set as a completely generic mathematical tool for
modeling uncertainties. There is no limited condition to the description of objects.
Thus researchers can choose the form of parameters they need. This simplifies the
decision-making process and makes the process more efficient in the absence of
partial information.
A soft set can be considered as an approximate description of an object precisely
consisting of two parts, namely, predicate and approximate value set. Exact solu-
tions to the mathematical models are needed in classical mathematics. If the model
is so complicated that we cannot get an exact solution, we can derive an approx-
imate solution and there are many methods for this. On the other hand, in soft set
theory as the initial description of object itself is of an approximate nature, we need
not have to introduce the concept of an exact solution.
vii
viii Foreword
Soft set theory has rich potential for application in many directions, some of
which are reported by Molodtsov in his work. He successfully applied soft set
theory in areas such as the smoothness of functions, game theory, operation
research, Riemann integration, and elsewhere. Later he presented some definitions
on soft sets as a subset, the complement of a soft set and discussed in detail the
application of soft theory in decision-making problems. Applications have been
made to decision-making, business competitive capacity information systems,
classification of natural textures, optimization problems, data analysis, similarity
measures, algebraic structures of soft sets, soft matrix theory, parameter reduction
in soft set theory, classification of natural textures, and soft sets and their relation to
rough and fuzzy sets.
The book, Soft Sets: Theory and Applications, by Prof. S. J. John is a strong
contribution to the development of soft set theory. It examines the algebraic and
topological structure of soft sets. It also considers some hybrid structures of soft
sets. The book contains interesting applications to decision-making, medical and
financial diagnosis problems. It is my hope that researchers will apply the concepts
of soft set theory to the existential problem of climate change and related problems
such as world hunger, coronavirus, modern slavery, and human trafficking.
Creighton University, USA John N. Mordeson

June 2020
Preface
A challenging problem faced by researchers while working towards efficient

computational systems is the need to model inherent uncertainty, imprecision and
partial information in the computational problem itself. The methods developed in
this context, broadly known as reasoning under uncertainty or approximate rea-
soning or imprecise reasoning, bring about a significant paradigm shift, which
reflected the remarkable reasoning ability of the human mind for information
processing and analysis which is often better than present day computers. In this
context only the idea of Soft computing emerged whose guiding principle is
exploiting tolerance for imprecision, uncertainty, partial truth, and approximation to
achieve tractability, robustness and low solution cost. Traditionally, the major
ingredients of Soft Computing include fuzzy logic, neuro-computing and proba-
bilistic reasoning. More recently we have genetic algorithms, belief networks,
chaotic systems and many more added up.
As the role model for this type of reasoning is the human mind and the fact that
human reasoning is not crisp and admits degrees. Obviously, fuzzy logic and fuzzy
set theory introduced by Zadeh play a major role in soft computing. Owing to the
fact that there are limitations of fuzzy set theory such as non-clarity in assigning
membership values, lack of parameterization techniques, soft techniques should
have the freedom to draw ideas from other generalizations of sets like rough sets,
multisets, soft sets, etc., and potential hybridization among these. Among these,
rough set theory is well developed and applications of rough sets in soft computing
are well explored. But other structures like multisets, soft sets, fuzzy multisets,
fuzzy soft sets, etc., are still in developing stages and their applications in various
soft computing scenarios need to be explored.
The development of soft set theory by Molodtsov in 1999 is significant in this
context. Taking the advantage of parameterization technique, soft set theory
evolved as a powerful tool for decision making in information systems, data mining
and reasoning from data, especially when uncertainty is involved. Before the
effective utilization of any mathematical technique for real life applications, it is
desirable to have a strong theoretical support of the developed concepts. Apart from
the basic operations and notions that are relevant in the context, structure studies
ix
x Preface
involving algebraic, topological and lattice theoretic concepts together with pos-
sible hybridization of the novel concept with already existing well established
techniques are also most relevant. This book is a humble attempt towards consol-
idating all these in the context of soft sets. For this, this monograph relies heavily
on many published works of the author, doctoral thesis of author’s students and
works of many other colleagues and researchers in this newly emerging area.
As such, this book contains 6 chapters covering various aspects of soft sets from
theoretical to application problems. Apart from that, a brief historic development of
soft sets and related structures together with some future directions in the devel-
opment of soft set theory and applications is also provided.
Chapter 1 introduces the basic definitions and notions of soft structure. Tabular
representation, operations and many results including analogue of DeMorgan laws
and results involving Cartesian product, relations and functions are provided. The
notions of distance, similarity and entropy also form a part of this chapter. Chapter
concludes with the representation of fuzzy sets, rough sets and topological spaces as
particular types of soft set, justifying the fact that soft set is a generalized tool.
With the intention of enriching the theoretical studies, the algebraic structures of
soft sets are studied in Chap. 2. They include soft groups, normalistic soft groups,
soft BCK/BCI algebras, soft rings and modules and soft lattices.
Topology is a major branch of mathematics with many applications in the fields
of physical and computer sciences. Topological structures on soft sets are more
generalized methods and they can be useful for measuring the similarities and
dissimilarities between the objects in a universe which are soft sets. Chapter 3
discusses two different approaches to soft topology. The basic difference in these
approaches is that one of them considers a subcollection of a set of all soft sets in an
initial universe with a fixed set of parameters and the other one considers a sub-
collection from the set of all soft subsets of a given soft set in a universe. In this
chapter, both approaches are considered with respect to some standard typical
topological notions.
Category theory brings together various branches of mathematics into a united
whole and paves the way to describe and compare objects with similar and different
properties. Chapter 4 is an attempt to accommodate categorical concepts in the
context of soft sets and soft graphs. Further, the relationship between soft sets and
classical information systems is also explored.
A usual practice in applications of uncertainty modelling problems is the
hybridization of existing structures with the intention that the evolving hybrid
structure will have advantages of the constituent ones. Soft sets are also not an
exception and there are many hybrid structures involving soft sets which yielded
better results. Chapter 5 gives a panoramic view of these structures. They include
hybridization including fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets,
rough sets, etc.
In order to justify the relevance and importance of the introduced concepts in
various application scenarios and to show the relationships of soft sets with other
related fields, Chap. 6 discusses various applications of soft sets in many real
Preface xi
problems like decision making, parameter reduction, game theory and studies
involving incomplete data.
The book is primarily designed for scientists, researchers and students working
in the field of soft sets and other related areas like rough sets, fuzzy sets, graph
structures and hybrid models involving them. I sincerely hope that this book will
certainly be an important source for graduate and postgraduate students, teachers
and researchers in colleges/universities in various fields of engineering as well as
mathematics/physics. I believe that with the help of the global reputed nature of the
publisher, the cutting edge ideas consolidated in this book will find ways to create a
stimulating atmosphere for further active development of soft computing techniques
round the globe.
Calicut, India Sunil Jacob John

July 2020
Acknowledgements
I would like to express my deep sense of gratitude to all my co-authors, and

collaborators for their help, suggestion and active participation in developing a
book on Soft sets. Special thanks are due to Prof. Janusz Kacprzyk, the
Editor-in-chief of the book series for the encouragement and the support provided.
I am indebted to Prof. Sivaji Chakravorti, Director, National Institute of
Technology Calicut, India for providing the facilities and kind understanding. My
colleagues at the Department of Mathematics were always a source of inspiration
while the preparation of this title. I sincerely hope that the foreword to the title
provided by Prof. John N. Mordeson will add up to the readability and the
reachability of the book and I am greatly indebted to him.
The timely help and support provided by my postgraduate and doctoral students
were really remarkable. The patience and love of my family members, in particular
my wife Jinta and son Sujin is gratefully appreciated. I am most appreciative of Dr.
Leontina Di Cecco, Ms. Jayarani Premkumar and other personnel of Springer
Nature for their help during the preparation of this book.
xiii
Contents
Part I Historical Perspective of Soft Sets

1 Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Basic Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Tabular Representation of a Soft Set . . . . . . . . . . . . . . . . . 4
1.2 Operations of Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 De Morgan Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Cartesian Product, Relations and Partitions . . . . . . . . . . . . . . . . . . 13
1.3.1 Soft Set Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Induced Relations from Universal Set
and the Attribute Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Equivalence Relations and Partitions on Soft Sets . . . . . . . 15
1.3.4 Kernels of Soft Set Relations . . . . . . . . . . . . . . . . . . . . . . 18
1.3.5 Closures of Soft Set Relations . . . . . . . . . . . . . . . . . . . . . 20
1.3.6 Orderings on Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4 Soft Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 Distance and Similarity Measures . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.1 Similarity Measure of Two Soft Sets . . . . . . . . . . . . . . . . 28
1.5.2 Distances Between Soft Sets . . . . . . . . . . . . . . . . . . . . . . 29
1.5.3 Distance Based Similarity Measure of Soft Sets . . . . . . . . . 31
1.6 Softness of Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.7 Representations of Fuzzy Sets, Rough Sets and Topological
Spaces as Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.7.1 Fuzzy Sets as Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.7.2 Topological Spaces as Soft Sets . . . . . . . . . . . . . . . . . . . . 35
1.7.3 Rough Sets as Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 36
xv
xvi Contents
2 Algebraic Structures of Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1 Soft Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 Normalistic Soft Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 Soft BCK/BCI-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Soft Rings and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3.1 Idealistic Soft Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4 Soft Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.5 Soft Lattice Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.5.1 DeMorgan’s Laws in Soft Lattices . . . . . . . . . . . . . . . . . . 71
2.5.2 Properties of Soft Lattice Operations . . . . . . . . . . . . . . . . . 73
3 Topological Structures of Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1 Soft Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 Closure, Interior, Boundary and Limit Points . . . . . . . . . . . . . . . . 89
3.3 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4 Continuous Mappings and Connectedness . . . . . . . . . . . . . . . . . . 105
3.5 Another Approach to Soft Topology . . . . . . . . . . . . . . . . . . . . . . 108
3.5.1 Continuous Soft Set Functions . . . . . . . . . . . . . . . . . . . . . 110
3.5.2 Soft Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.6 Soft Topologies Generated by Soft Set Relations . . . . . . . . . . . . . 114
4 Soft Graphs, Soft Categories and Information Systems . . . . . . . . . . 117
4.1 Soft Relations and Equivalence Relations . . . . . . . . . . . . . . . . . . . 117
4.1.1 Soft Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1.2 Soft Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2 Soft Graphs and Chained Soft Graphs . . . . . . . . . . . . . . . . . . . . . 125
4.2.1 Soft Relation on a Soft Set . . . . . . . . . . . . . . . . . . . . . . . . 126
4.2.2 Soft Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.2.3 Operations on Soft Graphs . . . . . . . . . . . . . . . . . . . . . . . . 131
4.3 Chained Soft Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.4 Category of Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.4.1 Basic Notions of Category Theory . . . . . . . . . . . . . . . . . . 137
4.4.2 Objects in Category Sset(U) . . . . . . . . . . . . . . . . . . . . . . . 142
4.4.3 Morphisms in Sset(U) . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.4.4 General Properties of the Category of Soft Sets . . . . . . . . . 150
4.5 Category of Soft Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.5.1 General Properties of the Category of Soft Graphs,
SGr(U) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.5.2 Existence of an Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.6 Relationships of Soft Sets with Information Systems . . . . . . . . . . 167
5 Hybrid Structures Involving Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . 171
5.1 Fuzzy Soft Sets and Soft Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . 171
5.1.1 Fuzzy Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.1.2 Soft Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Contents xvii
5.2 Intuitionistic Fuzzy Soft Sets and Soft Intuitionistic

Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.2.1 Intuitionistic Fuzzy Soft Sets . . . . . . . . . . . . . . . . . . . . . . 178
5.2.2 Soft Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . 180
5.3 Hesitant Fuzzy Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.4 Soft Rough Sets and Rough Soft Sets . . . . . . . . . . . . . . . . . . . . . 186
5.4.1 Soft Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.4.2 Modified Soft Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . 191
5.4.3 Rough Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6 Applications and Future Directions of Research . . . . . . . . . . . . . . . . 195
6.1 Parameter Reduction and Decision Making Problems . . . . . . . . . . 195
6.1.1 A Decision Making Problem . . . . . . . . . . . . . . . . . . . . . . 196
6.1.2 Parameterization Reduction of Soft Sets . . . . . . . . . . . . . . 200
6.1.3 Parameter Reduction of Soft Sets by Means
of Attribute Reductions in Information Systems . . . . . . . . . 202
6.2 Medical and Financial Diagnosis Problems . . . . . . . . . . . . . . . . . 209
6.3 Soft Sets in Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.3.1 Two Person Soft Games . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.3.2 An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
6.4 Soft Matrix Theory and Decision Making . . . . . . . . . . . . . . . . . . 221
6.4.1 Soft Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.4.2 Products of Soft Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.4.3 Soft MaxMin Decision Making . . . . . . . . . . . . . . . . . . . 225
6.5 Soft Sets in Incomplete Information Systems . . . . . . . . . . . . . . . . 228
6.5.1 Algorithm for Data Filling . . . . . . . . . . . . . . . . . . . . . . . . 230
6.6 Future Directions of Research and Developments
in Soft Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
6.6.1 Background of Soft Computing Technology . . . . . . . . . . . 232
6.6.2 Current Status of Soft Set Theory Research . . . . . . . . . . . . 233
6.6.3 Some Future Directions of Research in Soft Set
Theory Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Acronyms
ar(R) Anti-reflexive kernel of R

Cov(U) Set of all coverings of the universe U
CS(U) Set of all covering type soft sets over U
F(XY) Family of all fuzzy relationships from X to Y
F(U) Set of all fuzzy subsets of U
FS(U) Set of all fuzzy soft sets over U
Fset Category of fuzzy sets
Grp Category of groups
HFE Hesitant fuzzy element
HFS Hesitant fuzzy set
e ðUÞ
H Set of all hesitant fuzzy sets in U
IFS(U) Intuitionistic fuzzy power set of U
PAS(U) Set of all Pawlak approximation spaces
Par(U) Set of all partitions of the universe U
P(U) Power set of U
PS(U) Set of all partition type soft sets over U
prðf A Þ Set of all parameter reductions of soft set ðF; AÞ
RED(P) Family of all reductions of P
Rþ Set of all non-negative real numbers
radM Jacobson radical of module
Set Category of sets
SGr(U) Category of soft graphs over U
S(P) Strength of path P
SM mn Set of all m n soft matrices over U
SMMDM Soft max-max decision making
SMmDM Soft max-min decision making
SmMDM Soft min-max decision making
SmmDM Soft min-min decision making
SSR(F,A) Collection of all soft set relations defined on (F,A)
Sset(U) Category of soft sets over U
xix
xx Acronyms
SL ðEÞ Family of all soft lattices over a lattice L with parameter set E
s(R) Symmetric kernel of R
SðUÞE Family of all soft sets over U with parameter set E
soc(M) Socle of module
Vec Category of vector spaces
Part I
Historical Perspective of Soft Sets
Introduction of fuzzy set theory by Zadeh [161] in 1965 by giving room for partial
membership for better handling practical situations made a paradigm shift in math-
ematics. The rationale behind this idea was the need of modelling imprecise human
knowledge. The diffusion of this concept to various applied sciences and industry
was rapid and Zadeh himself contributed much for this. This was followed by many
successful generalizations of fuzzy sets and literature is abundant on these. These
generalizations include L-fuzzy sets by Goguen [47], intuitionistic fuzzy sets by
Atanassov [11], type two fuzzy sets [163], bipolar fuzzy sets [166], hesitant fuzzy
sets by Torra [144], pythagorean fuzzy sets [159], picture fuzzy sets [35], spherical
fuzzy sets [71], fermatean fuzzy sets [123] and many more. Apart from these another
set of generalized structures useful for approximate reasoning was also developed
parallel mostly in a complementary manner to fuzzy set theory. They include rough
sets by Pawlak [98], multisets by Yager [158], genuine sets by Demirci et al. [37],
neutrosophic and plithogenic logic by Smarandache [121], multiple sets by Shijina
et al. [128, 129] etc.
Molodtsov developed soft set theory in a fundamentally different perspective.
The application of this theory can be used for meaningfully interpreting real life
problems in pure and applied sciences involving imprecise data. Current studies
shows that ambiguities in data mining problems can also be solved using soft set
theory techniques. The soft set theory could be used to interrogate and extend the
idea of probability, fuzzy set, rough set and intuitionistic fuzzy set further. The
disadvantage of lack of parameterization tool related to the concepts mentioned
above gave a higher realm to soft set theory. In short, unlimited nature of approximate
description is the greatest advantage of soft set theory.
While pondering over difficulties related to modelling uncertainties, eighteenth
century mathematicians identified probability theory as a solution, which addressed
uncertainty via randomness. The prominence of this was unchallenged till mid-
twentieth century. In 1965, Zadeh [161] introduced fuzzy sets for addressing impreci-
sion comprehensively. He expressed fuzziness via partial membership of an element
in a set. Basically a fuzzy set can be identified with a class fitted with an ordering
for elements which expresses the more or less belongingness of them in to that class
under consideration.
2 Historical Perspective of Soft Sets
In contrast to Aristotelian classical bi-valued logic, Polish mathematician Jan

Lukasiewicz (1878–1956) introduced three-valued logic. Lukasiewicz is regarded as
the main founder and contributor of multi-valued logic, which was later extended by
Zadeh to fuzzy or infinite valued logic. In fuzzy logic, reasoning of false and truth are
considered in a graded fashion but in classical logic absolutely true or false statements
only are considered. Fuzzy logic can be considered as a branch of multi-valued logic
based on the paradigm of inference under vagueness. Further, the introduction of
fuzzy sets led to the development of many hybrid mathematical structures also.
By employing the notion of an equivalence relation called indiscernibility relation,
Pawlak [98] in 1982 brought in theory of rough sets by means of lower and upper
approximations and boundary region of a set. Lower approximation consists of all
elements which surely belongs to the concepts and upper approximation consists of
all elements which possibly belongs to the concepts. The difference of lower and
upper approximations is the boundary region. If boundary is empty then set is crisp,
otherwise it is rough. One benefit of the rough set theory is that it does not require
any additional parameter or details regarding the data to extract information.
In 1994 Pawlak [101] published a paper titled “Hard and soft sets", in which he
used a unified approach by taking ideas from classical set theory, rough sets and
fuzzy sets for representing soft sets. Motivated by this work, D. Molodtsov [89] in
1999 published the paper titled “Soft set theory: first results", which is considered
as the origin of theory of soft sets. Apart from the basic notions of the theory, some
of its possible applications and some problems of the future research directions are
also discussed in this paper. This theory was further solidified by P. K. Maji et al.
[81] in 2003 by defining some fundamentals of the theory such as equality of two
soft sets, subset and super set of a soft set etc. As continuation of these ideas, many
extensions, hybridizations and extensions were put forward by many authors, some
of them are the following: Maji, Biswas and Roy [79] introduced fuzzy soft sets,
Wang, Li and Chen [148] introduced hesitant fuzzy soft sets and Pei and Miao [103]
explore the relationship between fuzzy soft sets and classical information systems.
The theory of soft sets is still developing very rapidly both in theoretical as well as
application perspectives.
The never ending probe of researchers for better and better modeling of uncer-
tainty, ambiguity and vagueness may add more and more structures similar to soft
sets which will more specifically and accurately solve many problems of real world.
The relevance of soft sets in this context is always worth mentioning.
Chapter 1
Soft Sets
The aim of introducing a soft structure over a set is to make a certain discretization
of such fundamental mathematical concepts with effectively continuous nature and
thus providing new tools for the use of the technology of mathematical analysis in
real applications involving uncertainty or imperfect data. This is achieved through
a certain parameterization of a given set. As usual, this new perspective of ideas
draw attention of both pure and applied mathematicians and researchers in many
related areas as well. Specifically, the specialists found the concept of a soft set
well coordinated with many other modern mathematical concepts such as fuzzy sets,
rough sets and many more. Further, this resulted in a series of works where soft
versions of mathematical concepts were realized.
1.1 Basic Definitions and Examples
A soft set gives an approximate description of an object under consideration in two

precise parts, namely predicate and approximate value set. Classical Mathematics
always need exact solutions to mathematical models. Increasing level of complexity
or complications in model makes it difficult to get exact solutions and one may go
for approximate solutions and there are many methods for this. On the other hand,
in soft set theory as the initial description of object itself is of approximate nature,
we need not have to introduce the concept of exact solution.
Soft set theory, which was introduced by Russian researcher Molodtsov [88] in
1999 is a completely generic mathematical tool for modeling uncertainties. There
is no condition imposed on the description of objects; so researchers can choose
any form of parameters they needed, which greatly simplifies the decision-making
process and make the process more efficient and reliable in the presence of partial
© The Editor(s) (if applicable) and The Author(s), under exclusive license 3
to Springer Nature Switzerland AG 2021
S. J. John, Soft Sets, Studies in Fuzziness and Soft Computing 400,
https://doi.org/10.1007/978-3-030-57654-7_1
4 1 Soft Sets
information. There are many techniques available for modeling real world complex
systems, such as the classical probability theory, fuzzy set theory introduced by Zadeh
[160], interval mathematics [61, 96, 135] etc. Major drawback of all these techniques
is the lack of parameterization of the tools and hence they could not be applied
successfully in tackling problems especially in areas like economics, environmental
and social sciences. Soft set theory is relatively free from the difficulties associated
with above mentioned techniques and has a wider scope for many applications in a
multidimensional way.
In this section basic definitions, an example and a tabular representation as intro-
duced by Molodtsov [88], Maji et al. [80], and Babitha and Sunil [15] are mentioned.
Definition 1.1 Let U be an initial universe set and E be a set of parameters. Let
P(U ) denotes the power set of U and A ⊂ E. A pair (F, A) is called a soft set over
U , where F is a mapping given by F : A → P(U ).
In other words, a soft set over U is a parameterized family of subsets of the universe
U . For ∈ A, F() may be considered as the set of -approximate elements of the
soft set (F, A).
Example 1 Let U be a set of all students under consideration. E is a set of parame-

ters. Each parameter can be a word or sentence. E = {brilliant, average, healthy}.
In this case, we can define a soft set (F, A) to point out the Nature of students
as follows: Suppose that there are six students in the universe U given by U =
{x1 , x2 , , x3 , x4 , x5 , x6 } and E = {e1 , e2 , e3 } where e1 stands for brilliant, e2 stands
for average and e3 stands for healthy. The soft set (F, A) where A = E defined as
F(e1 ) = {x1 , x2 , , x5 }, F(e2 ) = {x3 , x4 , x6 }, F(e3 ) = {x1 , x4 , x5 , x6 } gives the soft
set representing the nature of students. The soft set (F, E) is a parametrized family
{F(ei ) : i = 1, 2, 3} of subsets of the set U and gives us a collection of approximate
descriptions of an object.
Here note that for each e ∈ E, F(e) is a crisp set. So the soft set (F, A) is called
a standard soft set.
1.1.1 Tabular Representation of a Soft Set
For the purpose of storing a soft sets in computers, one may need the representation
in the form of a matrix or a table. The (i, j)th entry in table
of a soft set
1 if xi ∈ F(e j )
ti, j =
0 otherwise
With reference to Example 1 given above, the tabular representation of the soft set
is given Table 1.1.
1.2 Operations of Soft Sets 5
Table 1.1 Tabular Representation of the soft set in Example 1

U↓ e1 (brilliant) e2 (average) e3 (healthy)
x1 1 0 1
x2 1 0 0
x3 0 1 0
x4 0 1 1
x5 1 0 1
x6 0 1 1
1.2 Operations of Soft Sets
Various operations analogous to union, intersection, complement, difference etc. in

set theory will be discussed in the context of soft sets. Definitions and results given
in this section are due to [7, 15, 80, 123].
Definition 1.2 For two soft sets (F, A) and (G, B) over a common universe U , we
say that (F, A) is a soft subset of (G, B) if
(i) A ⊆ B, and
(ii) ∀ ∈ A, F() and G() are identical approximations.
We write (F, A)⊆ (G, B).
(F, A) is said to be a soft super set of (G, B), if (G, B) is a soft subset of (F, A).
We denote it by (F, A)⊇ (G, B)).
Two soft sets (F, A) and (G, B) over a common universe U are said to be soft
equal if (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (F, A).
Definition of soft subset can be modified with replacing condition (ii) in Definition
1.2 “F() and G() are identical approximations” by “F() ⊆ G()”.
Definition 1.3 Let E = {e1 , e2 , e3 , .., en } be a set of parameters. The NOT set of E
denoted by ¬E is defined by ¬E = {¬e1 , ¬e2 , ¬e3 , .., ¬en } where ¬ei = not ei ∀i.
Definition 1.4 The complement of a soft set (F, A) is denoted by (F, A)c and is
defined by (F, A)c = (F c , ¬A) where F c : ¬A → P(U ) is a mapping given by
F c (¬α) = U − F(α), ∀¬α ∈ ¬A.
We call F c to be the soft complement function of F. Clearly (F c )c is the same as
F and ((F, A)c )c = (F, A). It is also known as neg-complement as F c is defined on
the NOT set of the parameter set.
Definition 1.5 Let U be an initial universe set, E be the set of parameters, and
A ⊂ E.
(i) (F, A) is said to be a relative null soft set (with respect to the parameter set A),
denoted by A , if ∀ ∈ A, F() = φ, (null-set).
6 1 Soft Sets
(ii) (F, A) is said to be a relative whole soft set (with respect to the parameter set
A), denoted by UA , if ∀ ∈ A, F() = U .
E is called the null soft set
The relative null soft set with respect to E denoted by
over U .
E is called the absolute
The relative whole soft set with respect to E denoted by U
soft set over U .
Definition 1.6 The relative complement of a soft set (F, A) is denoted by (F, A)r
or (F, A) and is defined by (F, A)r = (F r , A) where F r : A → P(U ) is a mapping
given by F r (α) = U − F(α), ∀α ∈ A.
Clearly, we have the following propositions.
Proposition 1.1 If A and B are two sets of parameters then we have the following:
(i) ¬(¬A) = A
(ii) ¬(A ∪ B) = (¬A) ∪ (¬B)
(iii) ¬(A ∩ B) = (¬A) ∩ (¬B)
Proposition 1.2 Let U be a universe, E a set of parameters, A, B, C ⊂ E. If (F, A),
(G, B) and (H, C) are soft sets over U , Then
U
(i) (F, A)⊆ A .

(ii) A ⊆(F, A).
(F, A).
(iii) (F, A)⊆

(iv) (F, A)⊆(G, B), (G, B)⊆ (H, C) implies (F, A)⊆ (H, C).
(v) (F, A) = (G, B) and (G, B) = (H, C) implies (F, A) = (H, C).
Definition 1.7 The union of two soft sets (F, A) and (G, B) over the common
U is a soft set (H, C), where C = A ∪ B and for each e ∈ C,
universe ⎧
⎪
⎨ F(e), if e ∈ A − B
H (e) = G(e), if e ∈ B − A .
⎪
⎩
F(e) ∪ G(e), if e ∈ A ∩ B
We write (F, A)
∪(G, B) = (H, C),
Definition 1.8 The intersection of two soft sets (F, A) and (G, B) over the common
universe U is a soft set (H, C), where C = A ∩ B, and H (e) = F(e) ∩ G(e), ∀e ∈ C.
We write (F, A)
∩(G, B) = (H, C).
Definition 1.9 Let (F, A) and (G, B) be soft sets over a common universe U
such that A ∩ B = φ. Then the restricted union of (F, A) and (G, B) denoted by
(F, A) ∪ R (G, B) and is defined as (F, A) ∪ R (G, B) = (H, C) where C = A ∩ B
and for all c ∈ C, H (c) = F(c) ∪ G(c).
Definition 1.10 Extended intersection of two soft sets (F, A) and (G, B) over the
⎧ A) ∩ E (G, B) and is the soft set (H, C), where
common universe U , denoted by (F,
⎪
⎨ F(e), if e ∈ A − B
C = A ∪ B, and ∀e ∈ C, H (e) = G(e), if e ∈ B − A .
⎪
⎩
F(e) ∩ G(e), if e ∈ A ∩ B
Definition 1.11 Let (F, A) and (G, B) be soft sets over a common universe U such
that A ∩ B = φ. Then the restricted difference of (F, A) and (G, B) denoted by
(F, A) R̃ (G, B) and is defined as (F, A) R̃ (G, B) = (H, C) where C = A ∩ B and
∀c ∈ C, H (c) = F(c) − G(c), the difference of the sets F(c) and H (c).
Definition 1.12 If (F, A) and (G, B) are soft sets over a common universe U , then
(F, A)AN D(G, B) denoted by (F, A) ∧ (G, B) is defined as (F, A) ∧ (G, B) =
(H, A × B) where H (a, b) = F(a) ∩ G(b) for every (a, b) ∈ A × B.
Definition 1.13 If (F, A) and (G, B) are soft sets over a common universe U ,
then (F, A)O R(G, B) denoted by (F, A) ∨ (G, B) is defined as (F, A) ∨ (G, B) =
(K , A × B) where K (a, b) = F(a) ∪ G(b) for every (a, b) ∈ A × B.
For soft sets (F, A), (G, B) and (H, C) over the same universe U with A, B, C
subsets of the parameter set E, the following theorems hold:
Theorem 1.1 Properties of union operation
(a) (F, A)∪((G, B)∪(H, C)) = ((F, A) ∪(G, B))∪(H, C)
(b) (F, A)∪UA = U
A , (F, A) E = U
∪U E , (F, A) A = (F, A)
∪
(c) (F, A) need not be a soft subset of (F, A) (G, B), then
∪(G, B). But if (F, A)⊂
(F, A)
(F, A)⊂ ∪(G, B), moreover (F, A) = (F, A) ∪(G, B)
(d) (F, A)∪(G, A) = A and (G, A) =
A if and only if (F, A) = A
(e) (F, A)∪((G, B)∩(H, C)) = ((F, A) ∪(G, B))∩((F, A)
∪(H, C))
(f) ((F, A)
∩(G, B))∪(H, C) = ((F, A) ∪(H, C))∩((G, B)∪(H, C))
Proof Proof of (a), (b), (e) and (f) are straight forward and follows easily from
definitions.
⎧ A)
(c) Let (F, ∪(G, B) = (H, C) where C = A ∪ B and
⎨ F(e) if e ∈ A − B
H (e) = G(e) if e ∈ B − A .
⎩
F(e) ∪ G(e) if e ∈ A ∩ B
It is obvious that if e ∈ A ∩ B, then H (e) = F(e) ∪ G(e), thus F(e) and H (e)
need not be the same approximations. Thus (F, A) need not be a soft subset of
(F, A) ∪(G, B).
Now let (F, A)⊂ (G, B). Then, it is clear that A ⊂ A ∪ B = A. We need to show
that F(e) and H (e) are the same approximations for all e ∈ A. Let e ∈ A, then
e ∈ A ∩ B = A, since A ⊂ B implies A − B = φ. Thus, H (e) = F(e) ∪ G(e) =
F(e) ∪ F(e) = F(e), as G(e) and F(e) are the same approximations for all e ∈ A.
This follows that H and F are the same set-valued mapping for all e ∈ A, as required.
(d) Suppose that (F, A) ∪(G, A) = (H, A), where H (x) = F(x) ∪ G(x) for all
x ∈ A. Since (H, A) = A from the assumption, H (x) = F(x) ∪ G(x) = φ ⇔
F(x) = φ and G(x) = φ ⇔ (F, A) = A and (G, A) = A for all x ∈ A. Now

assume that (F, A) = A and (G, A) = A and (F, A)∪(G, A) = (H, A). Since
F(x) = φ and G(x) = φ for all x ∈ A, H (x) = F(x) ∪ G(x) = φ for all x ∈ A.
Therefore, (F, A) ∪(G, A) = A.
Theorem 1.2 Properties of restricted union operation

8 1 Soft Sets
(a) (F, A) ∪ R (G, B) ∪ R (H, C) = (F, A) ∪ R (G, B) ∪ R (H, C).

(b) (F, A) ∪ R UA = U A , (F, A) ∪ R U A , (F, A) ∪ R
E = U A = (F, A).,
E = (F, A)
(F, A) ∪ R
(c) (F, A) (F, A) ∪ R (G, B), in general. But if (F, A)⊂ (G, B), then (F, A)⊂

(F, A) ∪ R (G, B), moreover (F, A) = (F, A) ∪ R (G, B)
(d) (F, A) ∪ R (G, A) = A ⇔ (F, A) = A and (G, A) = A
(e) (F, A) ∪ R ((G, B) ∩(H, C)) = (F, A) ∪ R (G, B) ∩((F, A) ∪ R (H, C)
∩(G, B)) ∪R (H, C) = ((F, A) ∪R (H, C))
(f) ((F, A) ∩((G, B) ∪R (H, C))
(g) (F, A) ∪R ((G, B) ∩ E (H, C)) = (F, A) ∪R (G, B) ∩ E (F, A) ∪R (H, C)
(h) ((F, A) ∩ E (G, B)) ∪R (H, C) = (F, A) ∪R (H, C) ∩ E (G, B) ∪R (H, C)
Proof (a) First, we investigate the left-hand side of the equality. Suppose that
(G, B) ∪ R (H, C) = (T, B ∩ C), where T (x) = G(x) ∪ H (x) for all x ∈ B ∩ C =
φ. And assume (F, A) ∪R (T, B ∩ C) = (W, A ∩ (B ∩ C)), where W (x) = F(x) ∪
T (x) = F(x) ∪ (G(x) ∪ H (x)) for all x ∈ A ∩ (B ∩ C) = φ.
Now consider the right-hand side of the equality. Suppose that (F, A) ∪R
(G, B) = (M, A ∩ B), where M(x) = F(x) ∪ G(x) for all x ∈ A ∩ B = φ · And
let (M, A ∩ B) ∪R (H, C) = (N , (A ∩ B) ∩ C), where N (x) = M(x) ∪ H (x) =
(F(x) ∪ G(x)) ∪ H (x) for all x ∈ (A ∩ B) ∩ C = φ. Since W and N are the same
mapping for all x ∈ A ∩ (B ∩ C) = (A ∩ B) ∩ C, the proof is completed.
(b) Proof of (b) follows directly from the definitions.
(c) Since A A ∩ B without any extra condition being given, (F, A) (F, A)
∪R (G, B) in general. Now assume that (F, A) is a soft subset of (G, B) and
(F, A) ∪R (G, B) = (H, A ∩ B = C), where H (x) = F(x) ∪ G(x) for all x ∈ C.
Then,
(F, A)⊂ (G, B) ⇔ A ⊂ A ∩ B = A and F(e) and G(e) are the same approxima-
tions for all e ∈ A ⇔ H (e) = F(e) ∪ G(e) = F(e) ∪ F(e) = F(e) for all e ∈ A.
Thus, F and H are the same set-valued mapping for all e ∈ A, so the proof is com-
pleted.
(d) Proof follows from the fact that (F, A) ∪R (G, A) = (F, A) ∪ (G, A)) and
Theorem 1.1(d).
(e) First, we handle the left-hand side of the equality. Suppose that (G, B) ∩
(H, C) = (T, B ∩ C), where T (x) = G(x) ∩ H (x) for all x ∈ B ∩ C. Let (F, A)
∪R (T, B ∩ C) = (W, A ∩ (B ∩ C)), where W (x) = F(x) ∪ T (x) = F(x)
∪ (G(x) ∩ H (x)) for all x ∈ (A ∩ B) ∩ C.
Now consider the right-hand side of the equality. Assume that (F, A) ∪R (G, B) =
(M, A ∩ B), where M(x) = F(x) ∪ G(x) for all x ∈ A ∩ B = φ. And let (F, A) ∪R
(H, C) = (N , A ∩ C), where N (x) = F(x) ∪ H (x) for all x ∈ A ∩ C = φ. Sup-
pose that (M, A ∩ B) ∩(N , B ∩ C) = (K , (A ∩ B) ∩ (A ∩ C)) = (K , (A ∩ B)
∩ C), where K (x) = M(x) ∩ N (x) = (F(x) ∪ G(x)) ∩ (F(x) ∪ H (x)) = F(x)
∪(G(x) ∩ H (x)) for all x ∈ (A ∩ B) ∩ C. Since W and K are the same set-valued
mapping, the proof is completed.
(f) By similar techniques used to prove (e), (f) can be illustrated, and is therefore
omitted.
(g) Suppose that (G, B) ∩ E (H, C) = (T, B ∪ C), where
⎧
⎨ G(e) if e ∈ B − C
T (e) = H (e) if e ∈ C − B
⎩
G(e) ∩ H (e) if e ∈ B ∩ C
Assume that (F, A) ∪ R (T, B ∪ C) = (M, A ∩ (B ∪ C)), where M(x) = F(x) ∪
T (x) for all x ∈ A ∩ (B ∪ C). By taking into account the properties of operations in
set theory and the definitions of M along with T and considering that T is a piecewise
function, we⎧can write the below equalities for M:
⎨ F(e) ∪ G(e) if e ∈ A ∩ (B − C) = (A ∩ B) − (A ∩ C)
M(e) = F(e) ∪ H (e) if e ∈ A ∩ (C − B) = (A ∩ C) − (A ∩ B)
⎩
F(e) ∪ (G(e) ∩ H (e)) if e ∈ A ∩ (B ∩ C)
for all e ∈ A ∩ (B ∪ C).
Now consider the right-hand side of the equality. Suppose that (F, A) ∪R
(G, B) = (Q, A ∩ B), where Q(x) = F(x) ∪ G(x) for all x ∈ A ∩ B = φ. Assume
(F, A) ∪R (H, C) = (W, A ∩ C), where W (x) = F(x) ∪ H (x) for all x ∈ A ∩
C = φ. Let⎧ (Q, A ∩ B) ∩ E (W, A ∩ C) = (N , (A ∩ B) ∪ (A ∩ C)), where
⎨ Q(e) if e ∈ (A ∩ B) − (A ∩ C)
N (e) = W (e) if e ∈ (A ∩ C) − (A ∩ B)
⎩
Q(e) ∩ W (e) if e ∈ (A ∩ B) ∩ (A ∩ C) = A ∩ (B ∩ C)
for all x ∈ (A ∩ B) ∪ (A ∩ C). By taking into account the definitions of Q and W,
⎧ N as below:
we can rewrite
⎨ F(e) ∪ G(e) if e ∈ (A ∩ B) − (A ∩ C)
N (e) = F(e) ∪ H (e) if e ∈ (A ∩ C) − (A ∩ B)
⎩
(F(e) ∪ G(e)) ∩ (F(e) ∪ H (e)) if e ∈ A ∩ (B ∩ C)
This follows that N and M are the same set-valued mapping when considering the
properties of operations on set theory, which completes the proof.
(h) By similar techniques used to prove (g), (h) can be illustrated, and is therefore
omitted.
Similar theorems follow for extended intersection and intersection also. Proofs are
in similar lines and hence omitted.
Theorem 1.3 Properties of extended intersection operation
(a) (F, A) ∩ E ((G, B) ∩ E (H, C)) = ((F, A) ∩ E (G, B)) ∩ E (H, C)
A = (F, A), (F, A) ∩ E
(b) (F, A) ∩ E U A = A
(G, B), then (F, A) ∩ E
(c) (F, A) ∩ E (G, B)(G, B), in general. But if (F, A)⊂

(G, B)⊂(G, B), moreover (F, A) ∩ E (G, B) = (G, B)
(d) (F, A) ∩ E ((G, B) ∪R (H, C)) = ((F, A) ∩ E (G, B)) ∪R ((F, A) ∩ E (H, C))
(e) ((F, A) ∪R (G, B)) ∩ E (H, C) = ((F, A) ∩ E (H, C)) ∪R ((G, B)
∩ E (H, C))
Theorem 1.4 Properties of intersection operation
(a) (F, A)
∩((G, B)
∩(H, C)) =((F, A) ∩(G, B))∩(H, C)

(b) (F, A)∩U A = (F, A), (F, A)∩U E = (F, A), (F, A)
A =
∩ A , (F, A) ∩
E

= A
(c) (F, A)
∩(G, B)(F, A), in general. But if (F, A)⊂ (G, B), then (F, A)

∩(G, B)⊂(F, A) moreover (F, A)
∩(G, B) = (F, A).
10 1 Soft Sets
(d) (F, A)

∩ ((G, B) ∪R (H, C)) = ((F, A)
∩(G, B)) ∪R ((F, A)
∩(H, C))
(e) ((F, A) ∪R (G, B))
∩(H, C) = ((F, A)
∩(H, C)) ∪R ((G, B)
∩(H, C))
(f) (F, A)
∩((G, B)∪(H, C)) = ((F, A)
∩(G, B))
∪((F, A)
∩(H, C))
(g) ((F, A)
∪(G, B))∩(H, C) = ((F, A)
∩(H, C))
∪((G, B)∩(H, C))
(h) ∩ ((G, B) ∼R (H, C)) =((F, A)
(F, A) ∩(G, B)) ∼R ((F, A)
∩(H, C))
(i) ((F, A) ∼R (G, B))
∩(H, C) = ((F, A)
∩(H, C)) ∼R ((G, B)∩(H, C))
Proposition 1.3 Let (F, A) be a soft set over U . Then we have the following;
(i) (F, A)∪(F, A)r = (F, A) ∪ R (F, A)r = U A
(ii) (F, A) ∩ E (F, A)r = (F, A)
∩(F, A)r = A
E )r =
(iii) (U A )r =
E , (U A
Proof Obvious.
1.2.1 De Morgan Laws
In this sub section, we show that the following De Morgan’s type of results hold in
soft set theory for different types of union, intersection, complements, AND and OR
operations. Results given in this section are taken from Ali et al. [7] , Maji et al. [80]
and Sezgin et al. [123].
Let (F, A) and (G, B) be two soft sets over a common universe U . Then we have
the following:
Theorem 1.5 (De Morgan laws with respect to relative complement, restricted union
and intersection)
(a) [(F, A) ∪ R (G, B)]r = (F, A)r
∩(G, B)r

(b) [(F, A)∩(G, B)] = (F, A) ∪ R (G, B)r
r r
Proof (a) Let (F, A)∪ R (G, B) = (H, C) where H (c) = F(c) ∪ G(c) for all c ∈
C = A ∩ B = ∅. Since ((F, A) ∪ R (G, B))r = (H, C)r , by definition H r (c) = U −
[F(c) ∪ G(c)] = [U − F(c)] ∩ [U − G(c)] for all c ∈ C.
Now (F, A)r ∩(G, B)r = (F r , A)
∩(G r , B) = (K , C) where C = A ∩ B. So by
definition, we have,
K (c) = F r (c) ∩ G r (c)

.
= (U − F(c)) ∩ (U − G(c))
= H r (c)∀c ∈ C
Hence [(F, A) ∪ R (G, B)]r = (F, A)r ∩(G, B)r .

(b) Let (F, A)∩(G, B) = (H, C) where H (c) = F(c) ∩ G(c) for all c ∈ C =
A ∩ B = ∅. Since ((F, A) ∩(G, B))r = (H, C)r , by definition H r (c) = U − (F(c)
∩ G(c)) = [U − F(c)] ∪ [U − G(c)] for all c ∈ C. Now (F, A)r ∪ R (G, B)r =
(F r , A) ∪ R (G r , B) = (K , C) where C = A ∩ B. So by definition, we have K (c) =
F r (c) ∪ G r (c) = (U − F(c)) ∪ (U − G(c)) = H r (c) for all c ∈ C. Hence
[(F, A) ∩(G, B)]r = (F, A)r ∪ R (G, B)r .
Theorem 1.6 (De Morgan laws with respect to extended intersection, union and
neg-complement)
(a) [(F, A) ∩ E (G, B)]C = (F, A)C
∪(G, B)C
(b) [(F, A)
∪(G, B)] = (F, A) ∩ E (G, B)C
C C
Proof (a) Suppose that (F, A) ∩ E (G, B) = (H, A ∪ B). Then

((F, A) ∩ E (G, B))C = (H, A ∪ B)c = (H c , (A ∪ B)) = (H c , A∪B) where
H c (e) = U − H (e) for all
⎧ e ∈A∪B.
⎨ F(e), if e ∈ A − B
By definition, H (e) = G(e), if e ∈ B − A
⎩
F(e) ∩ G(e), if e ∈ A ∩ B
⎧
⎪
⎨U \F(e) = F (e), i f e ∈A−B
c
Thus we have H (e) = U \G(e) = G c (e), i f e ∈B−A

c
⎪
⎩
U \(F(e) ∩ G(e)) = F c (e) ∪ G c (e), i f e ∈A∩B
⎧ A)
Moreover, let (F, ∪(G, B)c = (F c , A)
∪ (G c , B) = (K , A∪B).
c
⎪
⎨ F (e), i f e ∈A−B
c
Then K (e) = G c (e), i f e ∈B−A

⎪
⎩ c
F (e) ∪ G c (e), i f e ∈A∩B
c
Since H and K are indeed the same set-valued mapping, we conclude that
((F, A) ∩ E (G, B))c = (F, A)c ∪(G, B)c as required.
(b) By using a similar technique, part (b) can be proved.
Theorem 1.7 (De Morgan laws with respect to AND, OR, and neg complement)
(a) [(F, A) ∨ (G, B)]C = (F, A)C ∧ (G, B)C

(b) [(F, A) ∧ (G, B)]C = (F, A)C ∨ (G, B)C
Proof (a) Suppose that (F, A) ∨ (G, B) = (O, A × B). Therefore,

((F, A) ∨ (G, B))c = (O, A
c × B) c
= (O c , (A × B)). Now
(F, A) ∧ (G, B) = F , A) ∧ G c , B)
c c
= (J, A×B), where J (x, y) = F c (x) ∩ G c (y)

= (J, (A × B)
Now, take (α, β) ∈(A × B). Therefore,
O c (α, β) = U − O(α, β)
= U − [F(α ∪ G(β)]
= [U − F(α)] ∩ [U − G(β)]
= F c (α) ∩ G c (β)
= J (α, β)
Thus O c and J are same. Hence, proved.
Theorem 1.8 (De Morgan laws with respect to AND, OR, and relative complement)
12 1 Soft Sets
(a) [(F, A) ∨ (G, B)]r = (F, A)r ∧ (G, B)r

(b) [(F, A) ∧ (G, B)]r = (F, A)r ∨ (G, B)r
Proof (a) Suppose that (F, A) ∨ (G, B) = (O, A × B). Therefore,

((F, A) ∨ (G, B))r = (O, A × B)r = (O r , A × B) . Now, (F, A)r ∧ (G, B)r =
(F r , A) ∧ (G r , B) , = (J, A × B), where J (x, y) = F r (x) ∩G r (y). Let (α, β) ∈
A × B. Then, O r (α, β) = U \O(α, β) = U \[F(α) ∪ G(β)]= [U \F(α)] ∩ [U \
G(β)]= F r (α) ∩ G r (β) = J (α, β). Since O r and J are indeed the same set-valued
mapping, ((F, A) ∨ (G, B))r = (F, A)r ∧ (G, B)r .
In most of the studies related to algebraic and topological structures, we often need
to handle indexed family of entities. Feng et al. [41] gives various union, intersection
and/or operations for indexed families of soft sets.
Definition 1.14 Let (Fi , Ai )i∈l be a nonempty family of soft sets over a common uni-
verse U. The union of these soft sets is defined to be the soft set (G, B) such that B =
i∈I Ai and, for all x ∈ B, G(x) = i∈I (x) Fi (x), where I (x) = {i ∈ I |x ∈ Ai }. In

this case, we write i∈I (Fi , Ai ) = (G, B).
Definition 1.15 Let (Fi , Ai )i∈I be a nonempty family of soft sets over a common
universe set U . The AN D- soft set
∧i∈I (Fi , Ai ) of these soft sets is defined to be the
soft set (H, B) such that B = i∈I Ai and H (x) = i∈I Fi (xi ) for all x = (xi )i∈I ∈
B.
Definition 1.16 Let (Fi , Ai )i∈I be a nonempty family of soft sets over a common
universe set U . The O R- soft set
∨i∈I (Fi , Ai ) of these soft sets is defined to be the soft
set (H, B) such that B = i∈I Ai and H (x) = i∈I Fi (xi ) for all x = (xi )i∈I ∈ B.
Note that, if Ai = A and Fi = F for all i ∈ I, then ∧i∈I (Fi , Ai ) (respectively,

∨i∈I (Fi , Ai )) is denoted by
∧i∈I (F, A) (respectively,
∨i∈l (F, A)). In this case,
iel Ai =
I
i∈l A means the direct power A .
Definition 1.17 The restricted union of a nonempty family of soft sets (Fi , Ai )i∈I
over a common universe set U is defined as the soft set (H, B) = Ri∈I (Fi , Ai )
where B = i∈I Ai = ∅ and H (x) = i∈I Fi (x) for all x ∈ B.
Definition 1.18 The extended intersection of a nonempty family of soft sets

(Fi , Ai )i∈I over a common universe set U is defined as the soft set (H, B) =
Ei∈I (Fi , Ai ) such that such that B = i∈I Ai and H (x) = i∈I (x) Fi (x), where
I (x) = {i ∈ I |x ∈ Ai } for all x ∈ B.
Definition 1.19 Let (Fi , Ai )i∈l be a nonempty family of soft sets over a common
universe U. The intersection of these soft sets is defined to be the soft set (G, B)
such that B = i∈I Ai = ∅ and, for all x ∈ B, G(x) = i∈I Fi (x). In this case, we
write i∈I (Fi , Ai ) = (G, B).
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In besieging a fortress, the object is to erect batteries on particular points as near as
possible to the place, and to render the communications to and between them safe.
For these purposes, a ditch is commenced at a distance from the fortress, and is
carried on in a slanting direction towards it, the laborers being protected by the earth
thrown up on the side next the place. When these approaches have been carried as
near as requisite, another ditch called a parallel is dug in front or even around the
fortress, batteries being constructed on its line where necessary. Sometimes another
parallel is made within the outer one. Along these ditches the cannon, ammunition,
troops, &c. are conveyed in comparative safety to the different batteries.
Nevertheless Bourmont displayed here his determination to leave

nothing to chance, the success of which could be assured by caution
in the previous arrangements. The largest ships with the first and
second divisions of troops on board, passed around the extremity of
the peninsula, and anchored opposite its southwestern side on which
it had been resolved that the first descent should be made; a
steamer and some brigs entered the bay east of Sidi Ferruch, and
took positions so as to command the shore and the neck of the
peninsula, over which they could pour a raking fire, in case an attack
should be made by the Algerine forces at the moment of
disembarkation. Some rounds of grape shot from the steamer
dispersed the Arabs who were collected on the shore of the bay; the
fire was returned from the batteries; but it had no other effect than to
wound a sailor on board the Breslau, and it ceased after a few
broadsides from the brigs.
By sunset the vessels were all anchored at their appointed positions,

and preparations were instantly commenced for the disembarkation.
The broad flat bottomed boats destined to carry the troops to the
shore were hoisted out; each was numbered, and to each was
assigned a particular part of the force, so arranged that the men
might on landing, instantly assume their relative positions in the
order of battle.
All things being ready, at three o'clock on the morning of the 14th of
June, the first brigade of the first division under General Berthezéne,
consisting of six thousand men, with eight pieces of artillery were on
their way to the shore, in boats towed by three steamers. They were
soon perceived by the Algerines, who commenced a fire on them
from their batteries; it however produced little or no effect, and was
soon silenced by the heavier shot from the steamers and brigs in the
eastern bay. At four the whole brigade was safely landed, and drawn
up on the south side of the peninsula near the shore battery, which
was instantly seized. In a few minutes more, the white flag of France
floated over the Torreta Chica; a guard was however placed at the
door of the Marabout, in order to show from the commencement, that
the religion of the inhabitants would be respected by the invaders.
By six o'clock the whole of the first and second divisions were landed
together with all the field artillery, and the Commander-in-chief of the
expedition was established in his head quarters near the Marabout,
from which he could overlook the scene of operations. General
Valazé had already traced a line of works across the neck of the
peninsula, and the men were laboring at the entrenchments; they
were however occasionally annoyed by shots from the batteries, and
it was determined immediately to commence the offensive. General
Poret de Morvan accordingly advanced from the peninsula at the
head of the first brigade, and having without difficulty turned the left
of the batteries, their defenders were driven from them at the point of
the bayonet; they were then pursued towards the encampment,
which was also after a short struggle abandoned, the whole African
force retreating in disorder towards the city.
This success cost the French about sixty men in killed and wounded;
two or three of their soldiers had been taken prisoners, but they were
found headless and horribly mutilated near the field of battle. The
loss of the Algerines is unknown, as those who fell were according to
the custom of the Arab warfare carried off. Nine pieces of artillery
and two small howitzers by which the batteries were defended, being
merely fixed on frames without wheels, remained in the hands of the
invaders.
While the first brigade was thus employed, the disembarkation of the
troops was prosecuted with increased activity, and as no farther
interruption was offered, the whole army and a considerable portion
of the artillery, ammunition and provisions were conveyed on shore
before night. It was not however the intention of the commanding
general immediately to advance upon Algiers; his object was to take
the city, and he was not disposed to lose the advantage of the
extraordinary preparations, which had been made in order to insure
its accomplishment. The third division of the fleet containing the
horses and heavy artillery had not arrived; unprotected by cavalry his
men would have been on their march exposed at each moment to
the sudden and impetuous attacks of the Arabs, and it would have
been needless to present himself before the fortresses which
surround the city, while unprovided with the means of reducing them.
He therefore determined to await the arrival of the vessels from
Palma, and in the mean time to devote all his efforts to the
fortification of the peninsula, so that it might serve as the depository
of his materiel during the advance of the army, and as a place of
retreat in case of unforeseen disaster. The first and second divisions
under Berthezéne and Loverdo were accordingly stationed on the
heights in front of the neck of the peninsula, from which the
Algerines had been expelled in the morning; in this position they
were secured by temporary batteries and by chevaux de frise of a
peculiar construction, capable of being easily transported and
speedily arranged for use. The third division under the Duke
D'Escars remained as a corps of reserve at Sidi Ferruch, where the
engineers, the general staff and the greater part of the non-
combatants of the expedition were also established. Some difficulties
were at first experienced from the limited supply of water, but they
were soon removed as it was found in abundance at the depth of a
few feet below the surface.
On the 15th, it was perceived that the Algerines had established their
camp about three miles in front of the advanced positions of the
French, at a place designated by the guides of the expedition as Sidi
Khalef; between the two armies lay an uninhabited tract, crossed by
small ravines, and overgrown with bushes, under cover of which the
Africans were enabled to approach the outposts of the invaders, and
thus to annoy them by desultory attacks. Each Arab horseman
brought behind him a foot soldier, armed with a long gun, in the use
of which those troops had been rendered very dexterous by constant
exercise; when they came near to the French lines, the sharp
shooter jumped from the horse and stationed himself behind some
bush, where he quietly awaited the opportunity of exercising his skill
upon the first unfortunate sentinel or straggler who should appear
within reach of his shot. In this manner a number of the French were
wounded, often mortally by their unseen foes; those who left the
lines in search of water or from other motives were frequently found
by their companions, without their heads and shockingly mangled.
As the Arabs were well acquainted with the paths, pursuit would
have been vain as well as dangerous, and the only effectual means
of checking their audacity was by a liberal employment of the
artillery.
The labors of the French were interrupted on the morning of the

16th, by a most violent gale of wind from the northwest,
accompanied by heavy rain. The waves soon rose to an alarming
height, threatening at every moment to overwhelm the vessels,
which lay wedged together in the bays; several of them were also
struck by lightning, and had one been set on fire nothing could have
prevented the destruction of the whole fleet. Fortunately at about
eleven o'clock, the wind shifted to the east and became more
moderate; the waves rapidly subsided, and it was found that only
trifling injuries had been sustained by the shipping. Admiral Duperré
however did not neglect the warning, and he immediately issued
orders that each transport vessel should sail for France as soon as
she had delivered her cargo; the greater part of the ships of war,
were at the same time commanded to put to sea, and to cruise at a
safe distance from the coast, leaving only such as were required to
protect the peninsula.
On the 17th and 18th, some of the vessels arrived from Palma
bringing a few horses and pieces of heavy artillery, but not enough to
warrant an advance of the army. On the 18th, four Arab Scheicks
appeared at the outposts, and having been conducted to the
commander of the expedition, they informed him that the Algerines
had received large reinforcements, and were about to attack him on
the succeeding day. Bourmont however paid no attention to their
declarations, and gave no orders in consequence of them, although
it was evident from the increase in the number of their tents that a
considerable addition had been made to the force of his enemies.
On the day after the French had effected their landing, all the
Algerine troops except those which were necessary to guard the city
and the fortifications in its vicinity, were collected under the Aga's
immediate command, at his camp of Sidi Khalef; on the morning of
the 18th, the contingent of Oran also arrived, accompanied by a
number of Arabs who had joined them on the way. Thus
strengthened, and encouraged by the inactivity of the French, which
he attributed probably to want of resolution, Ibrahim determined to
make a desperate attack upon their lines, calculating that if he could
succeed in throwing them into confusion, it would afterwards be easy
to destroy them in detail. For this purpose he divided his army into
two columns, which are supposed to have consisted of about twenty
thousand men each; the right column under Achmet Bey of
Constantina was destined to attack Loverdo's division, which
occupied the left or northern side of the French position; the other
column was to be led by Ibrahim in person, with Abderrahman Bey of
Tittery as his lieutenant, against the right division of the invaders,
under Berthezéne.
At day break on the morning of the 19th, the Algerines appeared

before the lines of the French, who were however found drawn up,
and ready to receive them; the attack was commenced by the Arab
cavalry and Moorish regular troops intermingled, who rushed forward
rending the air with their cries, and endeavored to throw down the
chevaux de frise. The French reserved their fire, until the assailants
were near, and then opening their batteries poured forth a shower of
grape shot, which made great havoc in the ranks of the Algerines.
Nothing daunted however, the Moors and Arabs continued to pull up,
and break down the chevaux de frise, until they had gained
entrances within the lines; the action was then continued hand to
hand, the keen sabre of the African opposed to the rigid bayonet of
the European. In this situation there was less inequality between the
parties engaged, and the issue of the combat became doubtful.
Berthezéne's division however repulsed its assailants, and kept them
at bay; that of Loverdo was wavering when Bourmont appeared on
the ground, followed by a part of the reserved corps. He soon
restored order in the ranks, and having formed Loverdo's division
together with the reserve into a close column, he ordered them to
advance against their opponents. Achmet's forces were immediately
driven into a ravine where the artillery of the French having been
brought to bear upon them, they were after a few ineffectual
attempts to regain the height, thrown into disorder. Ibrahim's men
seeing this also lost their courage, and the route of the Africans
became general. The French had on the field only seventeen horses
which were attached to the artillery; as the Algerines could not
therefore be pursued very closely they were enabled to form again in
front of their camp at Sidi Khalef; but they were likewise driven from
this position, and followed for some distance beyond it, where the
ground being less favorable for cavalry, great numbers of their men
fell into the power of the invaders. Bourmont had issued orders to
spare the prisoners, but his troops irritated at the barbarities which
had been so frequently committed on their companions, disregarded
the injunction and put to death nearly every Algerine whom they
could reach. A few Arabs who were made prisoners, on being asked
respecting the forces and intentions of their General, haughtily bade
the French to kill and not to question them. The number of French
slain in this engagement according to the official reports, amounted
to fifty-seven, and of wounded to four hundred and sixty-three; but
little reliance can be placed on the exactness of Bourmont's
published accounts, and there is good reason for supposing that his
loss was much more serious. The destruction of life among the
Algerines was very great; they also left their camp of four hundred
tents, together with a large supply of ammunition, sheep and camels,
in the hands of their enemies.
The results of this action were highly important to the French, and
indeed it rendered their success certain. The Arabs began to
disappear, and the Turkish and Moorish soldiers retreated to the city,
from which it was not easy to bring them again to the field;
symptoms of insurrection among the populace also manifested
themselves. In this situation, it has been considered possible that
had Bourmont advanced immediately upon Algiers, the Dey would
have found it necessary to capitulate; there was however no reason
to believe that the disaffection would extend to the garrisons of the
fortresses, and the city could not have been reduced while they held
out.
On the 23d the vessels from Palma began to come in; the horses
were immediately landed, and two small corps of cavalry were added
to the troops encamped at Sidi Khalef. The fortifications of the
peninsula were also by this time completed, a line of works fifteen
hundred yards in length, having been drawn across the neck, and
armed with twenty-four pieces of cannon; by this means the whole of
the land forces were rendered disposable, as two thousand men
principally taken from the equipage de ligne3 of the fleet, were
considered sufficient for the security of the place. The provisions, &c.
were all landed, and placed within the lines, in temporary buildings
which had been brought in detached pieces from France;
comfortable hospitals were likewise established there, together with
bakeries, butcheries, and even a printing office, from which the
Estafette d' Alger, a semi-official newspaper, was regularly issued.
The communications between Sidi Ferruch and the camp, were
facilitated by the construction of a military road, defended by
redoubts and blockhouses placed at short intervals on the way.
3 A certain number of young men are annually chosen by lot in France, for the supply
of the army and navy, in which they are required to serve eight years. Those intended
for the navy, are sent to the dockyards, where they are drilled as soldiers, and
instructed in marine exercises for some time before they are sent to sea. The crew of
each public vessel must contain a certain proportion of those soldier sailors, who are
termed the equipage de ligne.
The Algerines encouraged by the delay of the French, rallied and

made another attack upon them at Sidi Khalef early on the morning
of the 24th. On this occasion but few Arabs and Kabyles appeared,
and the action was sustained on the side of the Algerines, almost
entirely by the Turks, the Moorish regulars, and the militia of the city,
who had been at length induced to leave its walls. The assailants
were spread out on a very extended line, which was immediately
broken by the advance of the first division of the French army, with a
part of the second in close column. A few discharges of artillery
increased the confusion; the Algerines soon began to fly, and were
pursued to the foot of the last range of hills which separated them
from the city. On the summit of one of these heights, were the ruins
of the Star Fort, which had been some years before destroyed,
"because it commanded the Casauba, and consequently the city;" it
was however used as a powder magazine, and the Africans on their
retreat, fearing lest it should fall into the hands of the French, blew it
up. The loss of men in this affair was trifling on each side. The only
French officer dangerously wounded was Captain Amédée de
Bourmont, the second of four sons of the General who accompanied
him on the expedition; he received a ball in the head, while leading
his company of Grenadiers to drive a body of Turks from a garden in
which they had established themselves, and died on the 7th of July.
While this combat was going on, the remainder of the vessels from
Palma, nearly three hundred in number, entered the bay of Sidi
Ferruch. Their arrival determined Bourmont not to retire to his camp
at Sidi Khalef, but to establish his first and second divisions five
miles in advance of that spot, in the valley of Backshé-dere, so that
the road might be completed, and the heavy artillery be brought as
soon as landed to the immediate vicinity of the position on which it
was to be employed. The third division was distributed between the
main body and Sidi Ferruch, in order to protect the communications.
This advantage was however dearly purchased; for during the four
days passed in this situation, the French suffered greatly from the
Algerine sharp-shooters, posted above them on the heights, and
from two batteries which had been established on a point
commanding the camp. In this way Bourmont acknowledges that
seven hundred of his men were rendered unfit for duty within that
period; he does not say how many were killed.
The necessary arrangements having been completed, and several
battering pieces brought up to the rear of the French camp,
Bourmont put his forces in motion before day on the 29th of June.
Two brigades of d'Escar's division which had hitherto been little
employed, were ordered to advance to the left and turn the positions
of the Algerines on that side; on the right the same duty was to be
performed by a part of Berthezéne's division, while Loverdo was to
attack the enemy in the centre. They proceeded in silence, and
having gained the summits of the first eminences unperceived,
directed a terrible fire of artillery upon the Algerines, who having only
small arms to oppose to it were soon thrown into confusion and put
to flight. The Moors and Turks took refuge in the city and the
surrounding fortifications, while the Arabs and Kabyles escaped
along the seashore on the southeast, towards the interior of the
country.
The French had now only to choose their positions from investing
Algiers, which with all its defences lay before them. Besides the
Casauba and batteries of the city, they had to encounter four
fortresses. On the southeastern side near the sea, half a mile from
the walls was Fort Babazon, westward of which, and one mile
southward from the Casauba, was the Emperor's castle, presenting
the most formidable impediment to the approach of the invaders.
This castle was a mass of irregular brick buildings, disposed nearly
in a square, the circumference of which was about five hundred
yards. From the unevenness of the ground on which it was built, its
walls were in some places sixty feet high, in others not more than
twenty; they were six feet in thickness, and flanked by towers at the
angles, but unprotected by a ditch or any outworks, except a few
batteries which had been hastily thrown up on the side next the
enemy. In the centre rose a large round tower of great height and
strength, forming the keep or citadel, under which were the vaults
containing the powder. On its ramparts were mounted one hundred
and twenty large cannon, besides mortars and howitzers, and it was
defended by fifteen hundred Turks well acquainted with the use of
artillery, under the command of the Hasnagee or Treasurer who had
promised to die rather than surrender. As it overlooked the Casauba
and the whole city, it was clear that an enemy in possession of this
spot and provided with artillery, could soon reduce the place to dust;
but it was itself commanded in a like manner, by several heights
within the distance of a thousand yards, which were in the hands of
the French. The next fortress was the Sittit Akoleit or Fort of twenty-
four hours, half a mile north of the city; and lastly a work called the
English fort was erected on the seashore near Point Pescada, a
headland about one-third of the way between Algiers and Cape
Caxine. The object of the French was to reduce the Emperor's castle
as soon as possible, and in the mean time to confine the Algerines
within their walls as well as to prevent them from receiving succors.
For the latter purposes, it was necessary to extend their lines much
more than would have been compatible with safety, in presence of a
foe well acquainted with military science; trusting however to the
ignorance and fears of his enemies, Bourmont did not hesitate to
spread out his forces, even at the risk of having one of his wings cut
off by a sudden sortie. Loverdo in consequence established his
division on a height within five hundred yards of the Emperor's
castle; Berthezéne changed his position from the right to the centre,
occupying the sides of mount Boujereah the heights immediately
west of the city; while d'Escars on the extreme left, overlooked the
Sittit Akoleit, and the English fort. These positions were all taken
before two o'clock in the day.
On the right of Berthezéne's corps, was the country house in which

the foreign consuls were assembled under the flag of the United
States. As its situation gave it importance, General Achard who
commanded the second brigade determined to occupy it, and even
to erect a battery in front of it. Major Lee the Commander in Chief of
the consular garrison, formally protested against his doing either,
maintaining that the flag which waved over the spot rendered it
neutral ground. The French General did not seem much inclined to
yield to this reasoning; but when it was also alleged that the erection
of the battery would draw the fire of the Algerine forts upon the
house, in which a number of females were collected, as well as the
representatives of several nations friendly to France, he agreed to
dispense with the execution of that part of his order, but his soldiers
were quartered on the premises, and his officers received at the
table of the consuls. The latter were, as might have been expected,
polished and gallant men; the soldiers were very unruly, and by no
means merited the praises which have been bestowed on their
moderation and good conduct, in the despatches of their commander
and the accounts of the historians.
The night of the 29th passed without any attack on the lines of the
French. Before morning the engineers under Valazé had opened a
trench within five hundred yards of the Emperor's castle, and various
country houses situated in the vicinity of that fortress, were armed
with heavy pieces and converted into batteries. As soon as this was
perceived from the castle, a fire was opened upon the laborers; but
they were already too well protected by the works which had been
thrown up, and few of the balls took effect. A sortie was next made
by the garrison, and for a moment they succeeded in occupying the
house of the Swedish Consul, in which a French corps had been
stationed; they were however immediately driven out, and forced to
retire to their own walls.
In order to divert the attention of the Algerines during the progress of

the works, false attacks were made on their marine defences by the
ships of the French squadron. On the 1st of July Admiral Rosamel,
with a portion of the naval force, passed across the entrance of the
bay, and opened a fire on the batteries, which after some time was
returned. Not the slightest damage appears to have been received
by either party, the French keeping, as the Admiral says, "à grande
portée de canon," that is to say, nearly out of the reach of the fire of
the batteries; one bomb is stated to have fallen in the vicinity of
Rosamel's ship. The effect of this movement not answering the
expectations of the French, as it did not induce the Algerines to
suspend their fires on the investing force, it was determined that a
more formidable display should be made. Accordingly on the 3d,
Admiral Duperré made his appearance before the place, with seven
sail of the line, fifteen frigates, six bomb vessels, and two steamers.
The frigate Belloné which led the way, approached the batteries and
fired on them, as she passed with much gallantry; the other ships
kept farther off, and as they came opposite the Mole, retired beyond
the reach of the guns, where they continued for some hours, during
which each party poured tons of shot harmless into the sea. As the
Admiral states in his despatch, "none of his ships suffered any
apparent damage, or notable less of men," except from the usual
"bursting of a gun on board the Provence, by which ten were killed
and fifteen wounded."
The high character for courage and skill which Admiral Duperré has
acquired by his long and distinguished services, precludes the
possibility of imagining that there could have been any want of either
of those qualities on his part in this affair. Indeed he would have
been most blameable had he exposed his ships and men to the fire
of the fortresses which extend in front of Algiers, at a period when
the success of the expedition was certain. The "moral effect" of
which the Admiral speaks in his despatch, might have been
produced to an equal or greater extent, by the mere display of the
forces in the bay; the only physical result of the cannonade, was the
abandonment of some batteries, on Point Pescada, which were in
consequence occupied by d'Escar's forces. The whole attack if it
may be so termed, was probably only intended to repress any
feelings of jealousy which may have arisen in the minds of the naval
officers and men, by thus affording them at least an ostensible right
to share with the army the glory of reducing Algiers.
BAI.
Bai was the Egyptian term for the branch of the Palm-tree. Homer
says that one of Diomede's horses, Phœnix, was of a palm-color,
which is a bright red. It is therefore not improbable that our word bay
as applied to the color of horses, may boast as remote an origin as
the Egyptian Bai.
THE CLASSICS.
Amid the signs of the times in the present age—fruitful in change if

not of improvement,—we have observed with pain not only a
growing neglect of classical literature, but continued attempts on the
part of many who hold the public ear to cast contempt on those
studies which were once considered essential to the scholar and the
gentleman, which formed such minds as Bacon's and Milton's, and
which afforded the most delightful of occupations to the leisure of a
Newton and a Leibnitz. In every age there has been a class of men
who from a depravity of taste, or else a passion for singularity, have
maligned all that is ancient or venerable. And sometimes with a
strange perversity of purpose, we see men wasting their
opportunities in a mischievous ridicule of useful pursuits which they
might have advanced and illustrated to the benefit of themselves and
mankind. Thus the seventeenth century, deeply imbued as it was
with the spirit of classical inquiry and the love of ancient literature,
gave birth to a Scarron and a Cotton, of whom the latter particularly
was fitted for higher pursuits, and the former perhaps worthy of a
better fate. But if in a spirit of indulgence for misguided genius we
pardon the offence of their jest for its wit, and feel that in so doing we
are involuntarily paying that tribute which is due to talent even when
misapplied, let us beware of extending the same indulgence to those
who from ignorance undervalue pursuits which they cannot
appreciate, or to those who contemn like the fox in the fable, objects
which they have vainly sought to obtain, or worse than all, to those
who have no better motive for their censure than the wish to pilfer
without detection, from the rich stores of those whom they have
banished from the public eye, and driven from their rightful abodes in
public recollection by a course of systematised slander. It would
perhaps be unjust to say that the opposers of the ancient and
learned universities of England, who have chiefly wrought the evil
influence upon English literature to which we have been alluding,
belong all of them to one of these three classes, but that many of
them may be ranked with the last we cannot doubt, when we see
what things they often send forth to the world as their own, and this
too with an air of the greatest pretension. That some of these
persons were actuated by better motives we must admit when we
trace to its origin the history of this partially successful war against
classical studies. The two universities of Oxford and Cambridge,
those ancient abodes of learning, to a certain degree undoubtedly
deserved the reproach of lagging behind the march of mind, in
denying to modern literature the share of attention to which it was
justly entitled. Absorbed in explorations of the past, and wedded to
the love of antiquity in all their associations, they sought literature in
her earliest haunts, and delighted most in their olden walks, which
they loved for the very frequency with which they had trodden them.
The system of study which had trained so many of their sons to
eminence, seemed to them the best, and they were too slow in
moulding its forms to the progress of science. It was endeared to
them not only from the nature of its pursuits, but from past success,
and it was no mean ambition which stimulated their sons to tread in
the paths which a Bacon or a Clarendon, a Newton or a Locke, had
trodden before them. And yet a little reflection should have taught
them that if these glorious models of human excellence had left
science where they found it, their reputations had never existed. A
fierce opposition at length sprung up to a system of study so narrow
and exclusive,—the growing wants of education demanded a
university in London, which project was opposed by many of the
friends of the old institutions. The elements of a party thus formed,
were soon combined, and as the controversy waxed warmer, they
attacked not only the venerable temples of learning, but the very
study of the ancient languages itself, at first, perhaps, because the
most celebrated abodes of this species of literature were to be found
in the universities to which they had become inimical. Like every
other literary controversy for some time past in England, this
question connected itself with the party politics of the day, and thus
many changed sides on the literary, that they might be together on
the political question. Strange as it may seem, it has been for some
time a reproach against the English that the Tories would not
encourage the Whig literature, and vice versâ. No reader of the
British periodicals for the last twenty years can have failed to remark
this fact, which serves to account for the progress of the literary
heresy which has already done so much to degrade English
literature and to deprave the tastes of those who read only the
English language. We shall not pause to inquire further into the
effects produced by this illicit connexion between politics and
literature in England, although it presents a highly interesting subject
of inquiry, and one which must deeply occupy much of the attention
of the historian who may hope hereafter to give an accurate account
either of the political or literary condition of that country for many
years past. Neither is it our purpose to arraign at the bar of public
opinion those who have draggled the sacred "peplon" itself in the vile
mire of party politics, although we sincerely believe that they will
have a heavy account to settle with posterity for this unhallowed
connexion. We merely allude to it by way of pointing out one of the
causes of the heresy which we mean to combat, from the belief that
it is mischievous, and the more especially as it diverts public
attention from the particular want of American literature. Unhappily
our reading in this country is chiefly confined to the English novelists
and the periodicals of the day, from which we derive a contempt for
the lofty and venerable learning of antiquity, and a belief that instead
of too little, we bestow too much attention upon classical literature in
America! That the novelists and trash manufacturers of the reviews
should foster this opinion is not at all surprising, for they find their
account in it. And yet it stirs the bile within us when we see a paltry
novelist who cannot frame his tale without borrowing his plot, or
conduct his dialogue without theft, affect to despise the study of
those authors whom he robs without any other restraint than the fear
of detection; or when we hear them offer to substitute their
lucubrations for the writings of the great masters of antiquity—men
who put forth opinions upon the most difficult questions in moral or
physical science, and support them only by a dogmatism which
would look down all opposition and frown upon any inquiry into the
grounds of their doctrines, who, like Falstaff, will give no reasons for
their moral or political opinions, and yet insinuate by their air of
pretension that they are "plenty as blackberries"—sciolist novelists
who doubt what is believed by all the most intelligent of their race,
and believe what no other persons but themselves can be brought to
believe—men who insinuate their superiority over the great models
of the human race by affecting to despise whatever they have
offered to the public view and modestly intimating their reliance upon
their own superior resources. Problems in morals and politics which
have filled with doubts and difficulties the minds of Bacon or Locke,
of Montesquieu or Grotius, are now settled at a stroke of the pen by
our novelist philosophers. Nothing is more common than to see the
solution of some one of them by the dandy hero of some fashionable
novel, who, sauntering from the dance to the coterie of philosophers
in blue, solves the difficulty en passant, and fearing that this trifling
occupation of so mighty a genius may attract attention, then hastens
to divert public observation from his sage aphorism and impromptu
philosophy by flirting with his friend's wife or playing with his poodle.
The conception of a costume is the only occupation worthy of his
fancy, and the composition of a dish the only subject which he would
have the world to think capable of tasking his powers of attention
and reflection; and yet all the learning of all the schools is shamed by
the display of this literary faineant who acquired his knowledge
without study, whilst inspiration only can account for the wisdom with
which he is instinct. A nation has groaned through long centuries of
almost hopeless bondage—the clank of a people in chains is heard
from the Emerald isle—a cry of distress fills the air—a mighty orator,
an O'Connell, arises before them, filling the public mind with
agitation and pointing the way to revenge. In the energy of despair a
portion of the captives have broken their manacles—they rush to
liberate their fellows—the air is full of their cry for revenge—the
conclave of Europe's wisest statesmen is at fault—a king trembles
on his throne—and what, gentle reader, do you suppose is to be the
result of these mighty throes and convulsions? why, just nothing,
literally nothing at all. A Countess of Blessington surveys the scene
from afar; reclining on an Ottoman, beneath a cloud of aromatic
odors she recollects the subject of conversation at her last "soiree;"
the idea flits across her brain with a gentle pang as it flies, that the
energy of O'Connell is becoming exceedingly vulgar, and that the
convulsions of a revolution so near her would be extremely trying to
her nerves, not to mention those of Messrs. Bulwer and D'Israeli.
Her resolution is taken, and at spare intervals between morning visits
and soirees, she writes the "Repealers," which is at once to settle
the agitations of a kingdom, and annihilate O'Connell himself. She
has no sooner finished, than washing her hands "forty times in soap
and forty in alkali," she despatches the production to Mr. Bulwer, who
looking upon the work pronounces it good; and lo! the succeeding
number of the New Monthly shall teach you the wonderful virtues of
the moral medicaments which come from the Countess of
Blessington's specific against Irish agitation. But who is Mr. Bulwer
himself? for in this age so wonderful for accomplishing great ends by
little means, it has become necessary to know him. Why a literary
magician, a sprite of Endor, who by the potency of his charm
conjures up the spirits of the mighty dead. Evoked by him the
departed prophets arise. A Peter the Great, and a Bolingbroke, a
Pope and a Swift, not to mention others of somewhat lesser note,
come forth and speak at his command as once they spoke. The
departed oracles of English literature are no longer mute. But the
visits of the dead are of necessity short. They have no time now for
such chit-chat as some may suspect they have hazarded whilst
living. They come on a mission of importance which they have barely
time to accomplish. The hidden secrets of policy are to be revealed,
mightly oracles in philosophy and criticism are to be declared. Truths
fall like hailstones, and wit descends in showers. But lo! what figure
is that which stalks across the scene and comes to take his part in
this play of phantasmagoria with which we have just been
entertained. Does he belong to the land of shadows or the world of
reality? "Under which king, Bezonian, speak or die." It is an
impersonation of the mental and moral qualities of Mr. Edward Lytton
Bulwer himself, not a prophet—but more than a prophet. The "most
wonderful wonder of wonders." Pope and Swift are overpowered by
his wit. The star of Bolingbroke pales before the superior effulgence
of this luminary, and Peter the Great, mute in astonishment, stands
"erectis auribus" to catch the oracles of government which flow from
the godlike man. The scene changes—whither doth he go? He
seizes the reins of government, he retrieves the affairs of a mighty
empire by way of recreating a mind exhausted with the play of its
mighty passions, and then wearied with the amusement, he turns in
quest of other pursuits. The rule of an empire and the affairs of this
world are objects too petty for the employment of his mind; he looks
for some higher subject, and finds it in himself—the only subject in
creation vast enough to fill the capacity of his spirit. He communes
with the stars—he talks to the "TOEN," and the "TOEN" replies to
him, and finally, big with his mighty purpose he achieves the task of
writing "his confessions." And as my lord Peter concocted a dish
containing the essence of all things good to eat, so this book is full of
something that is exquisite from every department of thought. Such
are the books which have displaced the writings of the masters of
antiquity and the old household books of the English tongue. You
may not take up a review or periodical now-a-days, but it shall teach
you the folly of bestowing your time upon the study of the ancients,
now that their writings afford so much that is more worthy of
attention. Alas! that such should be the priesthood who administer
the rites in the temple of English literature—the money changer has
indeed entered the temple, when those who write for money come in
to expel all who have written for fame. How often does it happen
now-a-days that the writer of a bawdy novel, derives reputation
enough from that circumstance, to assume the chair of criticism, and
exposing a front of hardened libertinism to the scorn of the good and
the contempt of the wise, avails himself of his situation to frown
down every attempt to resuscitate our decaying literature, by the
introduction of better models, and to restore health to the public
taste, which this very censor has contributed to deprave? There is no
more common occupation with such a man than the correction of the
errors of the most illustrious statesmen and philosophers in
magazine articles of some six or eight pages; the French revolution
is the favorite theme of his lofty speculations, and Napoleon's the
only character which he will exert himself to draw. With how much of
the lofty contempt of a superior spirit does he speak of the labors of
a Bentley, a Porson, a Parr, or an Elmsley; of a Gessner, a Brunck, a
Heyne, a Schweihauser or a Wolffe. The anxious labors, for years, of
such men as those go for nothing with him—they serve only to excite
his scorn, or else afford him the favorite subjects of his ridicule. With
the ingratitude of a malignant spirit, or the coarseness of ignorance,
he reviles the self-denying students who may be truly said to have
renounced the world in their enthusiastic search after the buried lore
of antiquity—men who have paled before the midnight lamp in their
ceaseless efforts to penetrate the obscurity of the past—lonely
eremites, who feed the lamps that cast their dim light on the votive
offerings which antiquity has laid upon the altar of knowledge—men
who have dwelt apart from their race and denied themselves the
common pleasures of life, that they might without distraction restore
the decaying temple of ancient literature, and recover for the use of
their own and future generations, treasures which else had been
buried and forgotten; who have lived in the past until they have
imbibed its spirit, and return like travellers full of the wisdom of
unknown lands, and rich with the accumulated experience of past
ages to shower their treasures and their blessings upon the
ungrateful many who despise them for their labors and taunt them
for their gifts, that they too may learn what a thing it is to cast pearls
before swine; and who, superior to the unmerited scorn of this world,
and to all the temptations of its grovelling pleasures, meekly bear
their ill treatment with no other emotion than the fear that the benefits
thus painfully acquired and freely bestowed, may turn out to be coals
of fire which they have been heaping upon unthankful heads. And
are men who labor for such objects as these to be ridiculed as
looking to things too small, because they sojourned so long in the
gloom of past ages, that their optics have been enlarged to discern
not only the mouldering monument, but the smallest eft that crawls
upon it? Shall they be taunted because they have learned to live in
mute companionship with their books, and like the lonely prisoner,
love objects which to others may seem inconsiderable, but are
endeared to them by all the force of a long association, whose chain
is interwoven link by link with the memory of their past? And if, like
Old Mortality, they love to restore each mouldering monument, and
retrace every time-worn inscription that may serve to renew their
silent communion with the hallowed and dreamy past, surely the
occupation may be pardoned, if not for its uses to others, at least for
the quiet affection and sweet enthusiasm of the dream which it
serves to awaken in the mind which is busy in the employment. But
the utilitarian spirit of the present age is ever ready to measure the
value of these pursuits by that pecuniary standard which alone it
uses. What are their fruits? Will they move spinning jennies or propel
boats? are they known on 'Change? how do they stand in the prices
current, and in what way will they put money in the purse? Strangely
as this may sound in the ears of those who love knowledge for itself
and its spiritual uses, and absurd as these things would have
appeared to the literary world a century ago, we much fear that we
must return answers to them satisfactory, in part, at least, before we
can even obtain an attentive hearing to what we shall say of their
higher excellences. It is true that classical attainments are in few
instances the objects of pecuniary speculation, nor is it our purpose
to hold out temptations to literary simony to those who, insensible of
the peace which the love of knowledge sheds abroad in the human
heart, would hope to sell or purchase that precious gift, for mere
money. If this were the only end which the student had in view, we
should regret to see him perverting to unworthy purposes the sacred
means to higher ends. To such a man learning has no temptations to
offer, for its best rewards he can never obtain without a change of
heart. We can no more unite the love of knowledge and of Mammon
than serve the two masters spoken of in Scripture. It is the rare
excellency of this holy taste that it releases us from servitude to the
unworthy desires which are too apt to fill the minds of those who
have never known what it was to thirst after the waters of truth. It is
indeed the redeeming spirit of the human mind, which casts out the
evil passions by which it had been possessed and torn. But there is
a class of students burning for distinction and ambitious of eminence
rather than wisdom, to whom we would appeal under the hope that
in the pursuit of their own lesser ends they will cultivate tastes which
may serve to awaken them to the more precious uses of knowledge.

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Soft Sets: Theory and Applications

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Fuzzy Graph Theory 1st Edition Sunil Mathew

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System and Architecture Sunil Kumar Muttoo

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ISSN 1434-9922 ISSN 1860-0808 (electronic)

Creighton University, USA John N. Mordeson

A challenging problem faced by researchers while working towards efﬁcient

Calicut, India Sunil Jacob John

I would like to express my deep sense of gratitude to all my co-authors, and

Part I Historical Perspective of Soft Sets

2 Algebraic Structures of Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Intuitionistic Fuzzy Soft Sets and Soft Intuitionistic

ar(R) Anti-reflexive kernel of R

In contrast to Aristotelian classical bi-valued logic, Polish mathematician Jan

1.1 Basic Definitions and Examples

A soft set gives an approximate description of an object under consideration in two

Example 1 Let U be a set of all students under consideration. E is a set of parame-

1.1.1 Tabular Representation of a Soft Set

Table 1.1 Tabular Representation of the soft set in Example 1

1.2 Operations of Soft Sets

Various operations analogous to union, intersection, complement, difference etc. in

Theorem 1.2 Properties of restricted union operation

(a) (F, A) ∪ R (G, B) ∪ R (H, C) = (F, A) ∪ R (G, B) ∪ R (H, C).

(d) (F, A)

1.2.1 De Morgan Laws

K (c) = F r (c) ∩ G r (c)

Proof (a) Suppose that (F, A) ∩ E (G, B) = (H, A ∪ B). Then

Thus we have H (e) = U \G(e) = G c (e), i f e ∈B−A

Then K (e) = G c (e), i f e ∈B−A

(a) [(F, A) ∨ (G, B)]C = (F, A)C ∧ (G, B)C

Proof (a) Suppose that (F, A) ∨ (G, B) = (O, A × B). Therefore,

= (J, A×B), where J (x, y) = F c (x) ∩ G c (y)

(a) [(F, A) ∨ (G, B)]r = (F, A)r ∧ (G, B)r

Proof (a) Suppose that (F, A) ∨ (G, B) = (O, A × B). Therefore,

Note that, if Ai = A and Fi = F for all i ∈ I, then ∧i∈I (Fi , Ai ) (respectively,

Definition 1.18 The extended intersection of a nonempty family of soft sets

Nevertheless Bourmont displayed here his determination to leave

By sunset the vessels were all anchored at their appointed positions,

The labors of the French were interrupted on the morning of the

At day break on the morning of the 19th, the Algerines appeared

The Algerines encouraged by the delay of the French, rallied and

On the right of Berthezéne's corps, was the country house in which

In order to divert the attention of the Algerines during the progress of

Amid the signs of the times in the present age—fruitful in change if

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Thus we have H (e) = U \G(e) = G c (e), i f e ∈B−A

Then K (e) = G c (e), i f e ∈B−A

= (J, A×B), where J (x, y) = F c (x) ∩ G c (y)