Logical equivalences and implications for quantified statement, Implications, Negation
Module 2:- Fundamentals of Counting Theory
Syllabus
The Rule of Sum – Extension of Sum Rule
The Rule of Product - Extension of Product Rule
Permutations
Combinations
The Binomial Theorem (without proof)
Combination with Repetition
The Pigeon hole Principle
The Principle of Inclusion and Exclusion Theorem (Without Proof) - Generalization of the Principle
Derangements
Module 3:- Relations and Functions
Syllabus
Cartesian Product - Binary Relation
Function – domain, range-one to one function, Image- restriction
Properties of Relations- Reachability Relations, Reflexive Relations, Symmetric Relations, Transitive relations, Anti-symmetric Relations, Partial Order relations, Equivalence Relations, Irreflexive relations
Partially ordered Set – Hasse Diagram, Maximal-Minimal Element, Least upper bound (lub), Greatest Lower bound(glb) (Topological sorting Algorithm- excluded)
Equivalence Relations
Partitions - Equivalence Class
Lattice - Dual Lattice, Sub lattice, Properties of glb and lub, Properties of Lattice, Special Lattice, Complete Lattice, Bounded Lattice, Completed Lattice, Distributive Lattice
Module 4:- Generating Functions and Recurrence Relations
Syllabus
Generating Function - Definition and Examples, Calculation techniques, Exponential generating function
First-order linear recurrence relations with constant coefficients - homogeneous, non-homogeneous Solution
Second-order linear recurrence relations with constant coefficients, homogeneous, non-homogeneous Solution
Module 5:- Algebraic Structures
Syllabus
Algebraic system-properties- Homomorphism and Isomorphism
Semigroup and monoid - cyclic monoid, sub semigroup and sub monoid, Homomorphism and Isomorphism of Semigroup and monoids
Group- Elementary properties, subgroup, the symmetric group on three symbols, The direct product of two groups, Group Homomorphism, Isomorphism of groups, Cyclicgroup