UNIT 1:
Vedic mathematics :
History and salient features,
Vedic
mathematics formulas, 16 sutras and 13 sub sutras, terms and
operations, Beejank, Vinculum Operations, High speed
addition by using the concept of completing the whole and
superfast subtraction by Nikhilam Sutram. Multiplication by
Nikhilam Sutra, Anatyodarshkeyapi,
Urdhavtriyaghbhyamsutram,
The Foundations: Logic, Sets and Functions: Introduction to
set theory, mathematical logic, prepositions, prepositional
equivalences, predicates and quantifiers. Importance of
Quantifiers.
The Foundations: Logic, Sets and Functions: Sets,
set operations, fuzzy sets, functions, functions
for computer science, sequences and summations. Logic and
computation in ancient Indian texts.
Mathematical reasoning: Introduction to Methods of proof,
mathematical induction. Use of mathematical induction to
solve different problems. Importance of recursions in
computer science, scope of recursions, Recursive definitions,
recursive algorithms.
UNIT 2:
Combinatorics: The basics of counting, The sum rule, The
product rule, The Pigeonhole Principle, Permutations with
repetitions, Permutations without repetitions, Circular
Permutations.
Applications of combinations:Applications of
Combinatorics to solve Committee problems, word problems,
puzzle problems etc. Applications of Combinatorics to
understand Telephone numbering plan, understanding Internet
addresses, Advanced counting techniques, recurrence
relations, solving recurrence relations, algorithm design, Basic
understanding of complexities, basic problems of complexity
of algorithms.
UNIT 3:
Relations: Relation definition, Importance of relations in
computer science, Relations and their properties, Unary
relations, Binary relations, Ternary relations, n-ary relations
and their applications, closures of relations, equivalence
relations, partial ordering. Representing relations, relation
matrix, relation graph, composite relation.
Operations on
relations: union, intersection and join. Concepts of least
upper bond, Greatest lower bond, maximal element, minimal
element, Greatest element, Least element of a partially ordered
set, lattices, sub lattices, chains and antichains.
UNIT 4:
Graphs: Introduction to Graphs, Importance of graph theory in
computer science, Graph terminology, representing graphs,
graph types, graph models, and graph isomorphism.
Connectivity, Euler and Hamiltonian Paths, shortest path
problems, planar graphs, graph colouring, chromatic number,
Euler’s formula, kuratowski’s theorem. The four colour
problem, Applications of Graph Colouring, Introduction to
Trees, applications of trees, tree traversal, trees and sorting,
Spanning trees, minimum spanning trees.
UNIT 5:
Languages and Grammars: Introduction to Languages and
Grammars, solving problems for validity of statements
according to the grammar. Importance of Language theory in
Computer Science, Importance of Derivation trees, solving
problems of Derivation trees, Importance of Parsing, Phrase
Structure Grammars, Types of Phrase structure grammars.