Lecture log & Exercises

Note: the exercises listed below are not to be submitted for grading.  However, you are very welcome to talk to me about them, and you should certainly work through all of them.

• Lec 1: Mon, 20 Aug 2012: principle of induction, equivalence relation
• Lec 2: Wed, 22 Aug 2012: examples of equivalence relations, equivalence class, in-class quiz 0.1
• Lec 3: Fri, 24 Aug 2012: partition, example of rational numbers as equivalence classes, divisibility
• exercises: induction (Sec 1.1: 2, 4); equivalence relations: (Sec 1.1: 7, 9)
• Lec 4: Mon, 27 Aug 2012: greatest common divisor (gcd), example of Euclid's algorithm
• Example 1(iv) on p.16 can be done with:  http://www.algebra.com/algebra/homework/divisibility/gcd-euclidean-algorithm.solver
• Lec 5: Wed, 29 Aug 2012: Euclid's algorithm, Euclid's lemma
• exercises: Sec 1.3: 1(i)(iv), 3, 5, 7(i)(ii)
• Lec 6: Fri, 31 Aug 2012: least common multiple (lcm), prime numbers, fundamental theorem of arithmetic
• exercises: Sec 1.4: 5, 11
• Lec 7: Wed, 5 Sep 2012: Euler's phi function, Euler's Theorem, Fermat's Little Theorem
• exercises: Sec 1.7: 1, 4, 7
• Lec 8: Fri, 7 Sep 2012: public key cryptography (RSA encryption)
• Lec 9: Mon, 10 Sep 2012: definition and basic properties of congruence
• exercises: Sec 2.1: 2, 3, 5, 6
• Lec 10: Wed, 12 Sep 2012: congruence being an equivalence relation, congruence classes
• Lec 11: Fri, 14 Sep 2012:  principal representative of a congruence class solving linear congruence equations
• exercises: Sec 2.1: 9(iii), (iv), (v)
• Lec 12: Mon, 17 Sep 2012: techniques for solving linear congruences, multiplicative inverse, using Euclid's algorithm to find a multiplicative inverse
• exercises: Sec 2.1: 11(ii), 13(i), 15(i)(ii)
• Exam 1: Wed, 19 Sep 2012.
• Lec 13: Fri, 21 Sep 2012: Chinese remainder theorem, applications of modular arithmetic (bar codes, checking divisibility), definition of \mathbb{Z}_m and addition/multiplication in it.
• exercises: Sec 2.1: 17, 18
• Lec 14: Mon, 24 Sep 2012: properties of addition and multiplilcation in \mathbb{Z}_m, addition and multiplication tables of \mathbb{Z}_5
• exercises: Sec 2.4: 3(i),(iii), 5(ii),(vi)
• Lec 15: Wed, 26 Sep 2012: units and zero divisors in \mathbb{Z}_m, number of units in \mathbb{Z}_m is \varphi (m), solving linear equations in \mathbb{Z}_m, axioms of a ring
• exercises: Sec 2.4: 6, 9
• Lec 16: Fri, 28 Sep 2012: examples of rings (including \mathbb{Z}_m, M_{2 \times 2}(\mathbb{R}) and the path algebra of a quiver)
• Lec 17: Mon, 1 Oct 2012: the path algebra of a quiver (continued), zero divisors
• Lec 18: Wed, 3 Oct 2012: integral domains, fields
• exercises: Sec 2.5: 5, 6, 7, 12
• Lec 19: Fri, 5 Oct 2012: axioms of a group, examples of groups and their operation tables, abelian groups
• Lec 20: Mon, 8 Oct 2012: more examples of groups (of 2 by 2 and 3 by 3 matrices), \mathbb{U}_m (the multiplicative group of units in \mathbb{Z}_m)
• Lec 21: Wed, 10 Oct 2012: the Klein four group (the symmetry group of a rectangle that is not a square), and how it is isomorphic to \mathbb{Z}_2 \oplus \mathbb{Z}_2
• exercises: Sec 4.1: 2, 3, 10
• Lec 22: Fri, 12 Oct 2012: consequences of group axioms (Proposition 1, p.192), the dihedral group D_3 (the symmetry group of an equilateral triangle)
• exercises: Sec 4.1: 8, 12; Sec 4.2: 1, 2, 4
• Lec 23: Mon, 15 Oct 2012: subgroups, "subgroup criterion", proper/nonproper subgroups, trivial subgroup
• exercises: Sec 4.4: 2, 4
• Lec 24: Wed, 17 Oct 2012: subgroup criterion for finite groups, cyclic subgroups and generators, 
• exercises: Sec 4.4: 5, 12
• Lec 25: Fri, 19 Oct 2012: intersections of subgroups, subgroups generated by a set, order of an element 
• exercises: Sec 4.4: 14, 20
• Lec 26: Mon, 22 Oct 2012: direct products of groups, direct sums
• exercise: Sec 4.4: 26
• Exam 2: Wed, 24 Oct 2012 (BYOT)
• Lec 27: Fri, 26 Oct 2012: more on direct products/sums, operation-preserving functions
• exercises: Sec 4.4: 27
• Lec 28: Mon, 29 Oct 2012: examples and non-examples of operation-preserving functions, isomorphisms
• Lec 29: Wed, 31 Oct 2012: examples of isomorphisms of groups, properties of isomorphisms, classifying finite groups
• exercises: Sec 4.5: 1, 2
• Lec 30: Fri, 2 Nov 2012: homomorphisms of groups and examples
• exercises: Sec 4.5: 11, 12
• Lec 31: Mon, 5 Nov 2012: kernel and image of a group homomorphism, classification of cyclic groups
• Lec 32: Wed, 7 Nov 2012: definition of permutations, definition of permutation group S_n, isomorphism between S_3 and D_3 (the dihedral group that is the symmetry group of an equilateral triangle), different notations for permutations (including cycle notations)
• exercises: Sec 4.7: 1
• Lec 33: Fri, 9 Nov 2012: non-disjoint vs disjoint cycles, cycle decomposition of a permutation, every permutation is a product of transpositions (2-cycles)
• exercises: Sec 4.7: 2, 5, 6
• Lec 34: Mon, 12 Nov 2012: even and odd permutations
• exercises: Sec 4.7: 4, 8
• Lec 35: Wed, 14 Nov 2012: group homomorphism from (S_n,\circ) to (\mathbb{Z}_2,+), the n-th alternating group A_n
• Lec 36: Fri, 16 Nov 2012: Cayley's Theorem, isomorphism between the subgroup ({R_0, R_1, R_2},*) of the dihedral group (D_3,*) and the subgroup ({e,(1 2 3), (1 3 2)},\circ) of (S_3,\circ)
• exercises: Sec 4.7: 12, 13
• Lec 37: Mon, 26 Nov 2012: (in-class presentations and teaching evaluation)
• Lec 38: Wed, 28 Nov 2012: (Rubik's cube competition) Counting colourings of an equilateral triangle (Section 4.8: p.240), equivalent colourings.
• Lec 39: Fri, 30 Nov 2012: (in-class presentations) Task 1 on p.241, counting formula (at the end of p.242) applied to the equilateral triangle 
• Lec 40: Mon, 3 Dec 2012: (in-class presentations) Counting isomers of a benzene ring (Task 12, part 6 on p.245).
• exercises: Sec 4.8: 
• Task 2
• Task 9 (i), (ii)
• Task 10
• Exam 3: Wed, 5 Dec 2012.