Special Set Linear Algebra and Special Set Fuzzy Linear Algebra

469 Pages·2010·7.86 MB·English

by  W. B. Vasantha Kandasamy

Special Set Linear Algebra - Cover:Layout 1 10/21/2009 9:50 AM Page 1
SPECIAL SET LINEAR ALGEBRA
AND SPECIAL SET FUZZY
LINEAR ALGEBRA
W. B. Vasantha Kandasamy
e-mail:[email protected]
web: http://mat.iitm.ac.in/home/wbv/public_html/
www.vasantha.in
Florentin Smarandache
e-mail:[email protected]
K Ilanthenral
e-mail:[email protected]

Editura CuArt
2009
This book can be ordered in a paper bound reprint from:

Editura CuArt
Strada Mânastirii, nr. 7
Bl. 1C, sc. A, et. 3, ap. 13
Slatina, Judetul Olt, Romania
Tel: [phone]
Editor: Marinela Preoteasa

Peer reviewers:
Dr. Arnaud Martin, ENSIETA E3I2-EA3876,
2 Rue Francois Verny, 29806 Brest Cedex 9, France
Prof. Mihàly Bencze, Department of Mathematics
Áprily Lajos College, Bra(cid:2)ov, Romania
Prof. Nicolae Ivaschescu, Department of Mathematics,
Fratii Buzesti College, Craiova, Romania.

Copyright 2009 by Editura CuArt and authors
Cover Design and Layout by Kama Kandasamy

Many books can be downloaded from the following
Digital Library of Science:
http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm

ISBN-10: 1-59973-106-1
ISBN-13: 978-159-97310-6-3
EAN: 9781599731063

Standard Address Number: 297-5092
Printed in the Romania
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CONTENTS
Dedication 5
Preface 6
Chapter One
BASIC CONCEPTS 7

Chapter Two
SPECIAL SET VECTOR SPACES AND
FUZZY SPECIAL SET VECTOR SPACES
AND THEIR PROPERTIES 33

2.1 Special Set Vector Spaces and their Properties 33
2.2 Special Set Vector Bispaces and their Properties 66
2.3 Special Set Vector n-spaces 103
2.4 Special Set Fuzzy Vector Spaces 156

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Chapter Three
SPECIAL SEMIGROUP SET VECTOR SPACES AND
THEIR GENERALIZATIONS 207

3.1 Introduction to Semigroup Vector Spaces 207
3.2 Special Semigroup Set Vector Spaces and
Special Group Set Vector Spaces 213

Chapter Four
SPECIAL FUZZY SEMIGROUP SET VECTOR
SPACES AND THEIR GENERALIZATIONS 295

4.1 Special Fuzzy Semigroup Set Vector Spaces
and their properties 295
4.2 Special Semigroup set n-vector Spaces 313

Chapter Five
SUGGESTED PROBLEMS 423

FURTHER READING 457

INDEX 462

ABOUT THE AUTHORS 467

4
DEDICATION
(15-09-1909 to 03-02-1969)
We dedicate this book to late Thiru C.N.Annadurai (Former Chief
Minister of Tamil Nadu) for his Centenary Celebrations. He is fondly
remembered for legalizing self respect marriage, enforcing two
language policy and renaming Madras State as TamilNadu. Above all
he is known for having dedicated his rule to Thanthai Periyar.
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PREFACE

This book for the first time introduces the notion of special
set linear algebra and special set fuzzy linear algebra. This
is an extension of the book set linear algebra and set fuzzy
linear algebra. These algebraic structures basically exploit
only the set theoretic property, hence in applications one
can include a finite number of elements without affecting
the systems property. These new structures are not only
the most generalized structures but they can perform multi
task simultaneously; hence they would be of immense use
to computer scientists.
This book has five chapters. In chapter one the basic
concepts about set linear algebra is given in order to make
this book a self contained one. The notion of special set
linear algebra and their fuzzy analogue is introduced in
chapter two. In chapter three the notion of special set
semigroup linear algebra is introduced. The concept of
special set n- vector spaces, n greater than or equal to three
is defined and their fuzzy analogue is given in chapter
four. The probable applications are also mentioned.
The final chapter suggests 66 problems. Our thanks are
due to Dr. K. Kandasamy for proof-reading this book. We
also acknowledge our gratitude to Kama and Meena for
their help with corrections and layout.

W.B.VASANTHA KANDASAMY
FLORENTIN SMARANDACHE
K.ILANTHENRAL

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Chapter One
BASIC CONCEPTS
In this chapter we just introduce the notion of set linear algebra
and fuzzy set linear algebra. This is mainly introduced to make
this book a self contained one. For more refer [60].
DEFINITION 1.1:Let S be a set. V another set. We say V is a set
vector space over the set S if for all v (cid:2) V and for all s (cid:2) S; vs
and sv (cid:2) V.
Example 1.1: Let V = {1, 2, ... , (cid:3)} be the set of positive
integers. S = {2, 4, 6, ... , (cid:3)} the set of positive even integers. V
is a set vector space over S. This is clear for sv = vs (cid:2) V for all
s(cid:2) S and v (cid:2) V.
It is interesting to note that any two sets in general may not be a
set vector space over the other. Further even if V is a set vector
space over S then S in general need not be a set vector space
over V.
For from the above example 1.1 we see V is a set vector
space over S but S is also a set vector space over V for we see
for every s (cid:2) S and v (cid:2) V, vs = sv (cid:2) S. Hence the above
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example is both important and interesting as one set V is a set
vector space another set S and vice versa also hold good inspite
of the fact S (cid:4) V.
Now we illustrate the situation when the set V is a set vector
space over the set S. We see V is a set vector space over the set
S and S is not a set vector space over V.
Example 1.2: Let V = {Q+ the set of all positive rationals} and
S = {2, 4, 6, 8, … , (cid:3)}, the set of all even integers. It is easily
verified that V is a set vector space over S but S is not a set
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vector space over V, for (cid:2) V and 2 (cid:2) S but .2(cid:5)S. Hence
3 3
the claim.
Now we give some more examples so that the reader becomes
familiar with these concepts.
(cid:7)(cid:10)(cid:8)a b(cid:9)
Example 1.3: Let M (cid:11)(cid:12)(cid:13) (cid:14) a, b, c, d (cid:2) (set of all
2(cid:6)2 (cid:10)(cid:17)(cid:15)c d(cid:16)
positive integers together with zero)} be the set of 2 (cid:6) 2
matrices with entries from N. Take S = {0, 2, 4, 6, 8, … , (cid:3)}.
Clearly M is a set vector space over the set S.
2x2
Example 1.4: Let V = {Z+ (cid:6) Z+ (cid:6) Z+} such that Z+ = {set of
positive integers}. S = {2, 4, 6, …, (cid:3)}. Clearly V is a set vector
space over S.
Example 1.5: Let V = {Z+ (cid:6) Z+ (cid:6) Z+} such that Z+ is the set of
positive integers. S = {3, 6, 9, 12, … , (cid:3)}. V is a set vector
space over S.
Example 1.6: Let Z+ be the set of positive integers. pZ+ = S, p
any prime. Z+ is a set vector space over S.
Note: Even if p is replaced by any positive integer n still Z+ is a
set vector space over nZ+. Further nZ+is also a set vector space
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over Z+. This is a collection of special set vector spaces over the
set.
Example 1.7: Let Q[x] be the set of all polynomials with
coefficients from Q, the field of rationals. Let S = {0, 2, 4, …,
(cid:3)}. Q[x] is a set vector space over S. Further S is not a set
vector space over Q[x].
Thus we see all set vector spaces V over the set S need be such
that S is a set vector space over V.
Example 1.8: Let R be the set of reals. R is a set vector space
over the set S where S = {0, 1, 2, … , (cid:3)}. Clearly S is not a set
vector space over R.
Example 1.9: (cid:2) be the collection of all complex numbers. Let
Z+ (cid:18){0} = {0, 1, 2, … , (cid:3)} = S. (cid:2) is a set vector space over S
but S is not a set vector space over (cid:2).
At this point we propose a problem.
Characterize all set vector spaces V over the set S which are
such that S is a set vector space over the set V.
Clearly Z+(cid:18){0} = V is a set vector space over S = p Z+(cid:18){0}, p
any positive integer; need not necessarily be prime. Further S is
also a set vector space over V.
Example 1.10: Let V = Z = {0, 1, 2, … , 11} be the set of
12
integers modulo 12. Take S = {0, 2, 4, 6, 8, 10}. V is a set
vector space over S. For s (cid:2) S and v (cid:2) V, sv (cid:19) vs (mod 12).
Example 1.11: Let V = Z = {0, 1, 2, … , p – 1} be the set of
p
integers modulo p (p a prime). S = {1, 2, … , p – 1} be the set.
V is a set vector space over S.
In fact V in example 1.11 is a set vector space over any proper
subset of S also.
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