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CS 201: Mathematics for Computer Science - I

Sets, proofs: [Weeks 1]

Sets, relations, functions, countable and uncountable sets.
Proofs; Proofs by deduction, contrapositive and contradiction (diagonalization).
Proofs by induction
* All these proof techniques can be covered through examples. The rigorous notion of proof will be covered in CS 202 and can be skipped here.

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.

[Weeks 4]

Inclusion-exclusion, Pigeonhole principal, Ramsey's theorem

Partial order: [Week 5]

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

Graph theory: [Week 6 and 7]

Definitions, degree, paths, cycles, Hamiltonian path, Eulerian cycles.
cycles and acyclic graphs.
Trees, spanning tree, 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, cyclic structure of Z_p^*.
Definition of 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.

Applications of group theory: [Week 14]

Burnside lemma and generalization to Polya's theorem (use of group theory in combinatorics).
Other interesting applications if time permits.

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.