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# Discrete Mathematics

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New

According to New CBCS (CHOICE BASE CREDIT SYSTEM) Syllabus w.e.f. 2019-20

A Text Book of

# Discrete Mathematics (MTC 112)

Author: Prof. Parshuram Ahire

ISBN: 978-93-5016-442-6

Price: 230/-

Rs.230

Contents

1. Logic
1. Introduction
2. Propositional Logic
Proposition
Propositional Variable
Logical Connectives
Examples
3. Propositional Equivalences
Definitions
Logical Equivalences
Examples
Logical Equivalences involving Conditional and Biconditionals
Constructing New Logical Equivalences
Examples
4. Rules of Inference
4.1 Argument in Propositional Logic
Valid Argument
Fallacy
4.2 Rules of Inference for Propositional Logic
Validity of Arguments
4.3  Using Rules of Inference to Build Arguments
Examples
5. Predicates and Quantifiers
5.1 Predicates
n-Place Predicate (or n-ary Predicate)
Examples
5.2  Quantification and Quantifiers
Universal Quantifier
Existential Quantifier
Examples
Other Quantifiers
Logical Equivalences Involving Quantifiers
Negating Quantified Expressions
Examples
2. Lattices and Boolean Algebra
1.  Introduction
Definition
1.1  Types of Relations
Reflexive Relation
Symmetric Relation
Transitive Relation
Anti-Symmetric Relation
Inverse Relation
Equivalence Relation
Partial Order Relation
Examples
2. Digraphs of Relations
2.1 Matrix Representation and Composition of Relations
2.2 Composition of Relations
Examples
2.3 Transitive Closure and Warshall's Algorithm
2.4 Warshallâ€™s Algorithm
Examples
3. Poset (Partially Ordered Set)
3.1 Partially Ordered Set (or Poset)
Definition
Examples4
3.2 Comparibility
3.3 Totally Ordered Set
3.4 Hasse Diagram7
Examples
Maximal and Minimal Elements
Definitions

4. Lattices
Definition
Examples
Definitions

Complement of Element
Examples
5. Boolean Functions
5.1 Operations in Boolean Algebra
Examples
5.2  Boolean Expressions and Boolean Functions
Definitions
5.3 Boolean Expression

Examples
Some definitions
5.4 Identities of Boolean Algebra
Examples
6. Duality
Dual Function
Duality Principle
Examples
6.1 Definition of Boolean Algebra
Examples
7.  Representing Boolean Function
Examples
Definitions
7.1 Disjunctive Normal Form (DNF): [or Sum of Products Expansion]
Definition
Examples
Definition
7.2 Conjunctive Normal Form (CNF): [or Product-of-Sums Expansion]
Definition
Examples

3. Counting Principles
1. Introduction
2. Cardinality of Set
Definitions
Examples
3. Cartesian Product of Sets
4. Basics of Counting
4.1 The Product Rule [or Multiplication Principle]
Examples
4.2 The Sum Rule [or Addition Principle]
Examples
4.3 The Principle of Inclusion Exclusion
Examples
5. The Pigeonhole Principle
Examples
5.1 The Generalized Pigeonhole Principle
Examples
5.2 Applications of Pigeonhole Principle
Examples
6. Generalized Permutations and Combinations
6.1 Permutation with Repetition
6.2 Combinations with Repetition
6.3 Permutations with Indistinguishable Objects
Examples
6.4    Distributing Objects into Boxes    39

4. Recurrence Relations
1. Introduction
2. Recurrence Relation
Examples
3. Linear Recurrence Relations with Constant Coefficients
Definition
Examples
4. Homogeneous Solutions
Definition
Method to Obtain Homogeneous Solutions
Examples
5. Particular Solutions
Examples
6. Total Solutions
Examples    31

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