What You will Learn ?

Module 1

Mathematical logic - Basic connectives and truth table, Statements, Logical Connectives,
Tautology, Contradiction. Logical Equivalence - The Laws of Logic, The Principle of duality,
Substitution Rules . The implication - The Contrapositive, The Converse, The Inverse.
Logical Implication - Rules of Inference. The use of Quantifiers - Open Statement, Quantifier.
Logically Equivalent – Contrapositive, Converse , Inverse , Logical equivalences and
implications for quantified statement, Implications , Negation .

Module 2

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

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
and 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 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 system-properties- Homomorphism and Isomorphism. Semi group and monoid –
cyclic monoid , sub semi group and sub monoid, Homomorphism and Isomorphism of Semi
group and monoids. Group- Elementary properties, subgroup, symmetric group on three
symbols ,The direct product of two groups, Group Homomorphism, Isomorphism of groups,
Cyclic group . Right Cosets - Left Cosets . Lagrange's Theorem.