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Course Overview

Discrete mathematics, sometimes called finite mathematics, is the study of mathematical structure that are fundamentally discrete, in the sense of not supporting notion of continuity. A study of discrete sets has become more and more necessary because of many application of computer science and various areas of engineering. In computer science, discrete mathematics are useful to study or express objects or problems in computer algorithm and programming languages. For instance, to improve the efficiency of a computer programs, we need to study its logical structure, which involves a finite number of steps each requiring a certain amount of time. Using the theory of combinatory and graph theory, major areas of discrete mathematics we can do this. Therefore, a study of these areas would complement and improve the understanding of courses based on algorithm and problem solving.
This Course is designed to give basic concepts of Propositions, Predicates, Boolean Algebra, Logic Circuit, Sets, Relations, Functions, Combinatorics, Partitions and Distributions.

BLOCK 1: Elementary Logic

Unit 1 : Prepositional Calculus

Propositions
Logical Connectives
o Disjunction
o Conjunction
o Negation
o Conditional Connectives
o Precedence Rule
Logical Equivalence
Logical Quantifiers

Unit 2: Methods of Proof

What is a Proof?
Different Methods of Proof
Properties Common to Logic and Sets
Relations
Cartesian Product
Relations and their types o Properties of Relations
Functions
Types of Functions
Operations on Functions
o Direct Proof
o Indirect Proofs
o Counter Examples
Principle of Induction

Unit 3 : Boolean Algebra and Circuits

Boolean Algebras
Logic Circuits
Boolean Functions

BLOCK 2: Basic Combinatorics

Unit 1: Sets, Relations and Functions

Introducing Sets
Operations on Sets
o Basic Operations

Unit 2 : Combinatorics – An Introduction

Multiplication and Addition Principles Permutations
Permutations of Objects not Necessarily Distinct
Circular Permutations Combinations
Binomial Coefficients Combinatorial Probability

Unit 3 : Some More Counting Principles

Pigeonhole Principle Inclusion-Exclusion Principle Applications of Inclusion – Exclusion Application to Surjective
Functions
Application to Probability
Application to Derangements

Unit 4 : Partitions and Distributions

Integer Partitions
Distributions
Distinguishable Objects into Distinguishable Containers
Distinguishable Objects into Indistinguishable Containers
Indistinguishable Objects into
Distinguishable Containers
Indistinguishable Objects into I ndistinguishable Container

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