Unfortunately, no single book will do. The texts we will be following are as follows. The first two will be our main textbooks.
Wk 
Dates 
Syllabus to be covered 
1 
1 Jan  7 Jan 
Logistics, Naive set theory done right (Tuesday notes); Axiom of Choice, Equivalence relations, Involutions (Wednesday notes); Partial orders and Zorn's lemma, Cardinality (Thursday notes) 
2 
8 Jan  14 Jan 
Integers and Rationals (Tuesday notes), Complete induction and the pigeonhole principle(Wednesday notes), Permutations and Combinations (Thursday notes) 
3 
15 Jan  21 Jan 
Cycles in permutations (Tuesday notes), Ordinary generating functions (Wednesday notes), Combinatorics and O.G.Fs (Thursday notes) 
4 
22 Jan  28 Jan 
Exponential generating functions, Graphs and the handshaking lemma (Tuesday notes), Trails, walks, Adjacency matrices (Wednesday notes), Eulerian tours (Thursday notes) 
5 
29 Jan  4 Feb 
Symmetric differences, Trees > Covered by Siddhartha Gadgil (Tuesday notes (a rough sketch of my take on Siddhartha's material)), Planar graphs (Wednesday notes ), Fundamental theorem of arithmetic and Euclid's algorithm (Thursday notes ) 
6 
5 Feb  11 Feb 
Linear Diophantine equations, Least common multiple, Infinitude of primes (Tuesday notes), Modular arithmetic (Wednesday notes), Definitions and examples of Rings and Fields (Thursday notes) 
7 
12 Feb  18 Feb 
Units and zero divisors in Z/mZ, definition of Integral Domains (Tuesday notes), Fermat's and Euler's theorems (Wednesday notes), Wilson's theorem, Pythagorean primes using the Gaussian integer ring(Thursday notes) ,Midterm on 18 Feb at 2:00 pm (The syllabus is everything up to and including week 6, i.e., up to and including the definitions and examples of rings and fields.) 
8 
19 Feb  25 Feb 
Midterm week 
9 
26 Feb  4 March 
Frobenius property, a version of Euler theorem for squarefree integers, definition of Ring homomorphisms (Tuesday notes), Examples of ring homomorphisms and characteristic of a commutative ring (Wednesday notes), Frobenius endomorphism (Thursday notes) 
10 
5 March  11 March 
DiffeyHellman protocol, RSA, Chinese Remainder theorem (Tuesday notes), Products of rings and the Chinese Remainder theorem, Polynomial rings (Wednesday notes), Division theorem and Euclid's algorithm for polynomials (Thursday notes) 
11 
12 March  18 March 
Fundamental theorem of arithmetic for polynomials, congruences, and Diophantinelike equations for polynomials (Tuesday notes), Chinese Remainder theorem for polynomials (Wednesday notes), Fundamental theorem of symmetric polynomials and cubic equations (Thursday notes) 
12 
19 March  25 March 
Solving the quartic and the AbelRuffini theorem for quintics (Tuesday notes), The Dihedral group and Cyclic groups (Wednesday notes), Subgroups of cyclic groups, Cayley's theorem (Thursday notes) 
13 
26 March  1 April 

14 
2 April  8 April 
Wk 
Test to be held on 
Homework (subject to changes; please check regularly) 
2 
11 Jan 

4 
25 Jan 

6 
8 Feb 

7 
15 Feb (Note this !) 

10 
8 Mar 

11 
15 Mar (Note this!) 

13 
29 Mar 

14 
5 Apr (Note this) 