Instructor:  Arvind Ayyer 
Office:  X15 (new wing) 
Office hours:  Tuesday, 45pm 
Phone number:  (2293) 3215 
Email:  (First name) at iisc dot ac dot in 
Class Timings:  Mondays, Wednesdays and Fridays, 11:00 — 11:50am 
Classroom: 
G  21, OPB To join the course on MS Teams, use the code gmw03ii 
Textbooks: 

TAs and office hours: 
Email ids are in paranthesis and end with at iisc dot ac dot in, and offices are in the maths department.

Tutorials:  Tuesdays 9:00 — 9:50am, G01 and G21 
The date for the midterms and final will be announced later.
Here are the weights for the homework and exams.
All marks will be posted online on Teams.
week  date  sections  material covered  homework and other notes 
1  1/1  Class cancelled 
  
2/1    Class cancelled 
  
3/1  (T) 2.12.3  Peano axioms  No homework :)  
5/1  Class cancelled 
  
2  8/1  (T) 3.1  Basic axioms  2.2.1, 2.2.4, 2.2.6, 2.3.2, 2.3.4 3.1.1, 3.1.3, 3.1.7, 3.1.10 
9/1    Discussion 
  
10/1  (T) 3.3, 3.4  Relations and functions  3.4.1, 3.4.3, 3.4.4  
12/1  (T) 3.3, 3.6, 8.1, 8.3 
Cardinalities of sets, countability  3.3.2, 3.3.5, 3.6.7, 3.6.6 8.1.2, 8.1.6, 8.3.3 

3  15/1  (T) 8.4, 8.5  Axiom of choice, Zorn's lemma  8.4.1, 8.5.3, 8.5.11, 8.5.18 
16/1    Quiz 1 
  
17/1  (T) 4.1  4.4  Integers, rationals and gaps in them  4.1.3, 4.1.8, 4.2.6, 4.4.2(a)  
19/1  (B) 1.11.2 2.12.2 
Pigeonhole principles and mathematical induction  Chap. 1: 2, 4, 9, 13 Chap. 2: 3, 5, 8, 11 

4  22/1  (B) 3.13.3  Permutations, choices and the binomial theorem  Chap. 2: 14, 28 Chap. 3: 1, 3, 7, 15, 21 
23/1    Discussion 
  
24/1  (B) 4.3, 5.1, 5.3  Combinatorial identities, compositions and partitions  Chap. 4: 3, 5, 9, 18 Chap. 5: 1, 7, 11 

26/1  Holiday 
  
5  29/1  (B) 5.2, 6.1  Set partitions  Chap. 5: 2, 4(a), 5, 16 
30/1    Quiz 2 
  
31/1  (B) 7.17.2  Permutations by cycles  Chap. 6: 2, 5, 8, 14, 17, 18, 22  
2/2  (B) 8.1  InclusionExclusion formulas and Ordinary generating functions 
Chap. 7: 1, 9, 13 Chap. 8: 1, 2, 8, 9 

6  5/2  (B) 8.2  Exponential generating functions  Chap. 8: 3, 16, 19, 20 
6/2    Discussion 
  
7/2  (B) 9.1  Graph theory definitions and Eulerian tours  Chap. 9: 2, 8, 12, 16  
9/2  (B) 10.1  Trees and Cayley's formula  Chap. 10: 2, 5, 7, 10, 15  
7  12/2  (B) 10.4  Spanning trees and the matrixtree theorem  Chap. 10: Example 10.22, 16, 18(b, c), 20 
13/2    Quiz 3 
  
14/2  (B) 12.1  Planar graphs  Chap. 11: Example 12.4, Cor. 12.7, Prop. 12.8  
16/2    Exam week 
  
8  19/2  Midsemester exam 
  
20/2    Exam week 
  
21/2  Exam week 
  
23/2  Exam week 
  
9  26/2  (IR) 1.1  Unique factorization in integers  Chap. 1: 1, 6, 10, 11 
27/2    Discussion 
  
28/2  (IR) 1.11.2  Unique factorization in polynomial rings  Chap. 1: 7, 13, 19, 20  
1/3  (IR) 1.31.4  Principal ideal domains, Z[i] and Z[ω]  Chap. 1: 31, 34, 36, 37  
10  4/3  (IR) 2.1  Infinitude of primes and Dirichlet's theorem  Chap. 1: 33, 35 Chap. 2: 1, 4, 6, 16, 17 
5/3    Quiz 4 
  
6/3  (IR) 2.2  Arithmetic functions  Chap. 2: 9, 10, 11, 13, 15, 22  
8/3  Holiday 
  
11  11/3  (IR) 3.13.3  Congruences in Z  Chap. 3: 2, 3, 4, 8 
12/3    Discussion 
  
13/3  (IR) 3.33.4  Euler's theorem, Fermat's little theorem and Chinese Remainder Theorem  Chap. 3: 5, 12, 16, 23  
15/3  (IR) 4.1  Units in Z_{n}  Chap. 4: 2, 4, 5, 13, 14, 22  
12  18/3  (IR) 5.1  Quadratic residues  Chap. 5: 2, 3, 6, 14, 15 
19/3    Quiz 5 
  
20/3  (IR) 5.25.3  Law of quadratic reciprocity  Chap. 5: 1, 6, 7, 10, 12, 16(i)  
22/3  (DF) 1.11.2  Basic properties of groups  Sec. 1.1: 1, 5, 9, 12, 13, 20 Sec. 1.2: 4, 9, 10 

13  25/3  (DF) 1.31.6  Examples of groups, homomorphisms and isomorphisms  Sec. 1.2: 7, 18 Sec. 1.3: 4, 10, 11, 15 Sec. 1.6: 1, 2, 5, 11, 18 
26/3    Discussion 
  
27/3  (DF) 1.7, 2.1  Group actions Subgroups 
Sec. 1.7: 2, 6, 8, 11, 15 Sec. 2.1: 1, 3, 5, 9 

29/3  Holiday 
  
14  1/4  (DF) 2.22.3  Centralizers, normalizers and cyclic groups  Sec. 2.2: 2, 3, 5(a), 6(a), 10 Sec. 2.3: 2, 12, 16 
2/4    Quiz 6 
  
3/4  (DF) 2.3, 31  Subgroups of cyclic groups and quotient groups  Sec 2.3: 20, 24 Sec 3.1: 1, 2, 6, 9, 11 

5/4  (DF) 3.1  Quotient groups and cosets  Sec 3.1: 12, 14(a, b), 22(a), 24, 26(a, b), 36  
15  8/4  (DF) 3.2, 3.3  Lagrange's theorem and first isomorphism theorem  Sec 3.2: 4, 5, 6, 10, 15 
9/4    Discussion 
  
10/4  (DF) 4.14.3  Cayley's theorem and class equation  Sec 4.1: 1, 4, 9(a) Sec 4.2: 1, 2, 9, 11 Sec 4.3: 2, 4, 10 

19  22/4    Final Exam 