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Amity University
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The objectives of Discrete Mathematical Structures are:
•
To introduce a number of Discrete Mathematical Structures (DMS) found to be serving
as tools even today in the development of theoretical computer science.
•
Course focuses on of how Discrete Structures actually helped computer engineers to
solve problems occurred in the development of programming languages.
•
Also, course highlights the importance of discrete structures towards simulation of a
problem in computer science and engineering.
•
Introduction of a number of case studies involving
problems of Computer
Technology.
Outcomes of this course are:
•
A complete knowledge on various discrete structures available in literature.
•
Realization of some satisfaction of having learnt that discrete structures are indeed
useful in computer science and engineering and thereby concluding that no mistake has
been done in studying this course.
•
Gaining of some confidence on how to deal with problems which may arrive in
computer science and engineering in near future.
•
Above all, students who studied this course are found to be better equipped in a
relative sense as far as preparation for entrance examinations involving placement
opportunities.
What is Discrete Mathematics then?
•
Mathematics is broadly divided into two parts; (i) the continuous mathematics and
(ii) the discrete mathematics depending upon the presence or absence of the limiting
processes.
•
In the case of continuum Mathematics, there do exists some relationship / linkage
between various topics whereas Discrete Mathematics is concerned with study of
distinct, or different, or un-related topics of mathematics curriculum; it embraces

several topical areas of mathematics some of which go back to early stages of
mathematical development while others are more recent additions to the discipline. The
present course restricts only to introducing discrete structures which are being used
as tools in theoretical computer science.
•
A course on Discrete mathematics includes a number of topics such as study of sets,
functions and relations, matrix theory, algebra, Combinatorial principles and discrete
probability, graph theory, finite differences and recurrence relations, formal logic and
predicate calculus, proof techniques - mathematical induction, algorithmic thinking,
Matrices, Primes, factorization, greatest common divisor, residues and application to
cryptology, Boolean algebra; Permutations, combinations and partitions; Recurrence
relations and generating functions; Introduction to error-correcting codes; Formal
languages and grammars, finite state machines. linear programming etc. Also, few
computer science subjects such as finite automata languages, data structures, logic
design, algorithms and analysis were also viewed as a part of this course.
•
Because of the diversity of the topics, it is perhaps preferable to treat Discrete
Mathematics, simply as Mathematics that is necessary for decision making in noncontinuous situations. For these reasons, we advise students of CSE / ISE / MCA, TE
(Telecommunication Engineering) to study this course, as they needs to know the
procedure of communicating with a computer may be either as a designer, programmer,
or, at least a user.
•
Of course, in today’s situation, this is true for all, although we do not teach to students
of other branches of engineering. In some autonomous engineering colleges, DMS is
being offered s an elective.
Considering these view points, you are informed to
undertake a course on Discrete Mathematical Structures so that you will be able to
function as informed citizens of an increasingly technological society.
•
Also, Discrete Mathematics affords students, a new opportunity to experience success
and enjoyment in Mathematics classes. If you have encountered numerous difficulties
with computation and the complexities of Mathematics in the past, then may I say that
this course is soft and a study requires very few formal skills as prerequisites.

•
In case if you are discouraged by the routine aspects of learning Mathematics, Discrete
Structures provides you a unique opportunity to learn Mathematics in a much different
way than the one employed by your teachers previously. Above all, Discrete
Mathematics is vital, exciting, and no doubt is useful otherwise you would not have
been suggested to register for this course.
•
Further, Discrete Mathematics course serves as a gateway for a number of subjects in
computer science and engineering. With these motivations, here, we initiate a detailed
discussion on some of the topics: These include Basic set theory, Counting techniques,
Formal Logic and Predicate calculus, Relations and functions in CSE, Order relations,
Groups and Coding etc.
•
Before, continuing, let me mention the difference between Discrete Mathematics and
other Mathematics; consider a bag of apples and a piece of wire. In the former, the
apples sit apart discretely from each other while in the latter, the points on a wire spread
themselves continuously from one end to the other.
•
Thus, the numbers 0, 1, 2, 3, . . . are sufficient to handle DMS, where as a real variable
taking values continuously over a range of values is required to deal with continuum
Mathematics. Hence,
•
Discrete Mathematics + Limiting Processes = Continuum Mathematics.
•
Prescribed text book:
•
Discrete and Combinatorial Mathematics by R. P. Grimaldi, PHI publications, 5th
edition (2004).
•
Reference Books:
•
Discrete Mathematical Structures by Kenneth Rosen, Tata McGraw Hill Publications
•
Discrete Mathematical Structures by Kolman, Busby and Ross, PHI publications

Basic set theory
A set is a well defined collection of well defined distinct objects. A set is usually denoted
by using upper case letters like A, B, G, T, X etc. and arbitrary elements of the set are
denoted using lower case letters such as a, b, g, v etc.
Universal set: The set of all objects under some investigation is called as universe or
universal set, denoted by the symbol U.
Consider a set A. Let
x be an element of A. This we denote symbolically by
xÎ A .
On the other hand, if y is not an element of the set, the same is written as y Ï A . Thus,
it is clear that with respect to a set A, and an element of the universal set U, there are only
two types of relationship possible; (i) the element x under question is a member of the set
A or the element
x need not be a member of A.
This situation may well be described by using binary numbers 0 and 1.
indicate that the element in question is a member of the set A.
We set x : 1 to
The notation x : 0
means that the element under study is not a member of the set A. There are a number of
ways of do this task. (i) Writing the elements of a set within the braces. For example,
consider A = {dog, apple, dead body, 5, Dr. Abdul Kalam, rose}. Certainly, A qualifies as a
set.
(ii) A set may be explained by means of a statement where elements satisfying some
conditions. Consider V = { x | x is a Engineering College Affiliated to VTU, Belgaum}
(iii) Definition of a statement may be given by means of a statement like Z denotes the set
of all integers. Thus, Z = {. . . -2, -1, 0, 1, 2, 3,. . }.
A null set is a one not having any elements at all. It is denoted by the symbol { } or as
.
Give few examples of null set or empty set.
Compliment of a set: Let A be a set. The compliment of A is defined as a set containing
elements of the universe but not the elements of the set A. Thus, A = { x Î U | x Ï A}

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