

Mathematical Foundations of Computer Science
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Lecture Notes

Introduction to Combinatorics
Introduction, Basic Counting Principles, Sum Rule, Product Rule,
Permutation of Sets, Theorems on Permutation of Sets, Combination of Sets,
Counting Paradigms, Partition of a Set, Sampling Paradigm, Sampling
without Replacement, Sampling with Replacement, Selection, Worked out
examples, Reference.

Counting Principles
Identities Through Counting, Vandermonde's Identities, Stirling number
of second kind, Fat Sets, Pigeonhole Principle, Binomial Theorem and
their Extended Forms, Principle of Inclusion and Exclusion, Examples,
Further Examples, Counting through Recurrence Equations, Derangements.

Recurrence Equations, Generating Functions
Recurrence Relations, Solving Recurrence Relations, Linear Homogeneous
Recurrence Relations with Constant Coefficients and Distinct Roots,
Case of Repeated Roots, Non Homogeneous Recurrence Relation,
Generating Functions, Linear Recurrence Relations, Exponential
Generating Function, Catalan Numbers, Derangements.

Recurrence Relations
Iteration Method, Master Method, Guess and Verify Method, Constructive
Induction, Transformations, Generating Function Techniques,
Inhomogeneous Recurrences, Miscellaneous.

Sets and Relations I
Set Theory, Some Definitions, Subset, Power set, Union, &
Intersection, Properties of Sets, Ordered Pair, Relation, Inverse
Relation, Walk, Trail, Path & Cycle, Some Relations, Equivalence
Relation, Closure of Relation, Composition of Relation, Reference.

Problem Solving and Mathematical Modelling
Introduction, What is a problem?, Abstraction of a Problem,
Astavadhani, Stepwise Refinement, Mathematical Models  What and
Why?, Components of a Mathematical Model, The Mathematical Modeling
Process, Examples  One Way Streets, Examples  Cyclic Quaderilateral
Test, Conclusion, Exercises, Reference.

Sets and Relation II
Introduction, Cantor's Concept of a set, The Basis of a Intuititive
Set Theory, Principle of Extensionality, Principle of Abstraction,
Principle of Comprehension, Inclusion, Operation on Sets, The Algebra
of Sets, Exercise on Sets, Relations, Some Theorems on Relations,
Function, Images of Sets under Relations, Relations and Graphs,
Special Classes of Relations, Equivalence and Partitions, Exercises
on Relations and Functions.

Discrete Probability Theory

Recurrence Equations II

Mathematical Logic
Assignment Problems and Solutions
Quiz


