cs302

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a.k.a.

Discrete Mathematics

Discrete mathematical Structures

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WHY STUDY DISCRETE

MATHEMATICS? Excellent tool for improving reasoning and problem solving skills. Modern mathematics deals with sets and not simple numbers. Information is stored and manipulated by computing machines in

discrete fashion.

Even real numbers are covered in DM as they are represented asfloating point numbers.

Knowledge of DM is integral for efficiently understanding and usingconcepts like Data Structures Database Theory Automata Theory Compiler Design Operating Systems .. And many more!

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COMPONENTS

MATHEMATICALLOGIC SET THEORY BOOLEANALGEBRA MATRICES

NUMBER THEORY RELATIONS FUNCTIONS POSETS ANDLATTICES

COMBINATRONICS ABSTRACT AND LINEAR ALGEBRA GRAPH THEORY

TREES PROBABILITY LANGUAGE, GRAMMAR ANDAUTOMATA

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COMPONENTS





The science dealing with the methods of reasoning.All methods of verification of algorithms use notations and methods of logic.Mathematical logic is the foundation on which Artificial Intelligence is built.Crucial for studying behavior of gates and circuits in hardware design.

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COMPONENTS





A set is a collection of well defined objects.The language of set theory is used in every mathematical subject.Programmers use sets of different objects & ways of manipulatingthemThe foundation behind the field of Fuzzy Logic.

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COMPONENTS

MATHEMATICALLOGIC

SET THEORY BOOLEANALGEBRA

MATRICES




Binary logic deals with variables that take on two variables 0and 1 and with operations that assume logical meaning.The foundation behind design of digital circuits.Crucial for understanding and efficiently designing computersystems.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




Data is arranged in sets whose elements are indexed by one or moresubscripts.Matrices provide a means of storing large amounts of data such that the datacan be handled element wise and as a whole.Extensively used in AI fields like robotics.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




The part of Mathematics that seeks to investigate the properties ofintegers.Has applications in transmission, coding and manipulation of data.Extensively used in Cryptography.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




The part of mathematics that deals with objects and theircomparisons.While executing programs, comparisons are made and different tasksare performed based on results.Relational databases stores information in the form of relationsbetween objects and their attributes.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




Functions describe connections between two different sets.Foundation for the field of mathematical modeling.Extensively used in domain of recursive function theory.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




A POSET is a special type of a set and a Lattice is a special type of anOrdered set.Used in PERT and CPM techniques crucial for Software ProjectManagement.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




The mathematics of Counting, Selecting and Arranging objects.It includes the theory of Permutations and Combinations.Application of combinatronics play important role in Algorithmic Analysis.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




Algebra uses different mathematical structures and theirmanipulations. Important structures are Groups, Rings andFields.Groups are used in Automata theory and Formal languages.Very important in developing techniques that help in transmissionof data that help in detecting and correcting errors.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




A Graph is a set of points that are interconnected by a set oflines.Applications include syntactic analysis, fault detection and diagnosisin computers and minimal path problems.A very useful practical problem-solving tool in a variety of fields.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




A tree represents hierarchical relations between data items.A binary search tree is a very important search structure in many applications.Trees are used to organize information in database systems and to represent thesyntactic structure of source programs in compilers.Used to analyze electrical circuits and represent structure of mathematicalformulas.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




Branch of mathematics which deals with phenomena whoseoutcomes involve uncertainity.Probability is measured on a scale from 0 to 1.Has applications in coding theory and data transmission and errordetection / correction.

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COMPONENTS

MATHEMATICALLOGIC


MATRICES




Defines structure to a language over a specified set of symbols.Used for developing and analyzing programming languages.Has applications in robotics, AI, Compiler construction andknowledge engineering.

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Syllabus in Brief UNIT 1

SET THEORY COMBINATIONS

OPERATIONS RELATIONS

TYPES AND PRPERTIES

REPRESENTATION

LATTICES FUNCTIONS

CLASSIFICATION

OPERATIONS

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RECURRENCE RELATIONS AND ALGORITHMS

METHODS OF SOLVING

COUNTING TECHNIQUES

PIGEONHOLE PRINCIPLE

PERMUTATION AND COMBINATION

MATHEMATICAL INDUCTION

VARIANTS

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ALGEBRIC STRUCTURES

GROUPS

TYPES AND PROPERTIES

ORDERS

OPERAIONS

RINGS

PROPERTIES FIELDS

PROPERTIES

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LOGIC

BOOLEAN ALGEBRA

AXIOMS AND THEOREMS

BOOLEAN EXPRESSION SIMPLIFICATION

ALGEBRIC

KARNAUGH MAP

DIGITAL CIRCITS PROPOSITIONAL LOGIC

PREDICATE LOGIC

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GRAPHS

REPRESENTATION

TYPES

PATHS

TREES

BINARY TREE

TRAVERSAL

SPANNING TREES

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A B

C

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Basics Definition

Elements

Membership

Representation

Tabular or Roster (Listing) Method

Set Builder or Rule Method

Bit String Method Special Sets

R , Z , Z+ , N , C , Q

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Types of SetsFinite Sets

Infinite Sets

Null SetSingleton Set

Subset

SupersetProper Subset

Equal Sets

Universal Set

Comparable Set Disjoint Set

Power Set

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Operations on Sets

Union

Intersection

Complement Difference

Symmetric Difference

Cardinality

Venn Diagrams

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Multisets Representation

Cardinality

Operations

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)()()(

Laws)()()((5)

)()(Laws)()((4)

Laws(3)

Laws'(2)

ofLaw)1(

CABACBA

veDistributiCABACBA

CBACBAeAssociativCBACBA

ABBA

eCommutativABBABABA

sDemorganBABA

ComplementDoubleAA

Set Identities

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A)BA(A

LawsAbsorptionA)BA(A(10)

LawsDomination=A,UUA(9)

LawsInverseAA,UAA(8)

LawsIdentityAUA,AA(7)

LawsIdempotentAAA,AAA(6)

Set Identities

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