Notes of discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets.

The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deal with finite sets, particularly those areas relevant to the business.

FOR T1

Sets, Relation, and Function:-
Operations and Laws of Sets
Cartesian Products
Binary Relation
Partial Ordering Relation
Equivalence Relation. 

POSET and Lattices:-

Partial orderings
POSETS
Totally Ordered Set
Dual Order
Hasse Diagram
Lexicographic Ordering
Well-Ordering Theorem
Lattices and their properties
Bounded Lattices
Sub Lattices
Direct Products

Notes for T1:- Notes1, Notes2

FOR T2

Counting Techniques:-

Principles of Mathematical Induction: The Well-Ordering Principle
Recursive definition
The Division algorithm: Prime Numbers
The Greatest Common Divisor: Euclidean Algorithm
The Fundamental Theorem of Arithmetic
Basic counting techniques-inclusion and exclusion
Pigeon-hole principle

Propositional Logic:-

Syntax, Semantics, Validity, and Satisfiability

Basic Connectives and Truth Tables

Logical Equivalence: The Laws of Logic, Logical Implication, Rules of Inference

The use of Quantifiers. 

Recursive relation: book-1

Normal form: book-2

Handwritten Notes: note-1, note-2

FOR T3

Graph Theory

Degree

Handshaking Theorem and Applications

Types of Graphs

Sub Graphs and Isomorphisms

Walk, Paths, Eulerian and Hamiltonian Walks

Shortest Path Problems

Planar Graphs

Graph Colouring

Trees

Spanning Trees

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Group Theory

Algebraic Systems Examples and General Properties

Semi Group and Monoid

Group

Sub-Group

Cyclic-group

Cosets and Lagrange Theorem

Normal Sub-Groups

Quotient Groups

Rings

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Boolean Algebra

Minimization of Boolean Expressions:

• Algebra Method
• Karnaugh Map(K-Map)

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