CSE

Source: https://cse.iitk.ac.in/pages/CS201.html Parent: https://cse.iitk.ac.in/pages/ResearchAreasNew.html

CS 201: Mathematics for Computer Science - I

Credits: 3-1-0-10 ( 1 hour of tutorial every week)

Sets, proofs: [Weeks 1]

Sets, relations, functions, countable and uncountable sets.\ Proofs; Proofs by deduction, contrapositive and contradiction (diagonalization).\

Basic Counting: [Weeks 2]

Selection/combination, arrangements/permutation, rule of sum, rule of product.\ Binomial coefficients, identities, multinomial coefficients.\ Selection with repetition/distributions of objects into cells, distinguishable/indistinguishable objects.\ Combinatorial problems with restrictions

Generating functions: [Week 3]

Recurrence relations to solve combinatorial problems.\ Generating functions to solve recurrence.

Counting techniques: [Weeks 4]

Inclusion-exclusion, Pigeonhole principle, Ramsey's theorem

Partial order: [Week 5]

Equivalence relations, partitions, partial order, posets, chain/antichain.

Graph theory: [Week 6 and 7]

Definitions, degree, paths, cycles, Hamiltonian path, Eulerian cycles.\ cycles and acyclic graphs.\ Trees, spanning trees, networks.

Number theory: [Week 8 and 9]

Divisibility, primes, division theorem, Euclid's gcd/extended Euclid's algorithm, Unique factorization domain.\ Modular arithmetic, sums and products, Chinese remaindering, Mobius inversion.

RSA: [Week 10]

Fermat's little theorem, Euler's theorem.\ Application: RSA.

Finite fields: [Week 11]

Z_p, the cyclic structure of Z_p^*.\ Definition of the field as a generalization to F_p.\ Application: Polynomials over F_p, error correction.

Group theory: [Week 12 and 13]

Definitions, examples (Z_n and Z_n^*)\ cyclic and dihedral group, abelian groups.\ Subgroups, cosets, partition\ permutation group, transpositions, cycle representation\ symmetries as a group.

Optional topics: [If time permits]

Rings\ Applications Of Group Theory (Burnside lemma and generalization to Polya's theorem, use of group theory in combinatorics).\ Other interesting applications.\

Books:

1) Kenneth Rosen, Discrete mathematics and its applications.\ 2) Norman Biggs, Discrete mathematics.\ 3) Chung Liu, Introduction to combinatorial mathematics.\ 4) David Burton, Elementary number theory.