This is the course page for Interlake High School’s FM class, Winter 2016.
The course meets 4.5 hours per week.
We just finished the FM 4 textbook on Group Theory and are now working on the FM 6 textbook on Discrete Mathematics.
The course will aim to prepare the student to succeed in university mathematics, to enrich the student’s understanding of what mathematics is, and to prepare the student to succeed on the IB FM exam.
Sets, Relations, Groups Exam Date: Monday 3/21
Number Theory Exam Date: Monday, 4/25
Graph Theory Exam Date: Tuesday, 5/31
The Review Sets at the end of each text, your problem sets, and your lecture notes are good study materials for the exams.
PS1: (9) Due 2/12
PS2: (13) Due 2/18
C.2: 4, 6, 8
PS3: (13) Due 3/4
E.2: 2, 4, 5
F.1: 1, 4, 7
F.2: 2, 6
F.3: 1, 4
PS4: (5) Due 3/14
H: 1, 3, 6
PS5: (8) Due 3/21
I: 1gh, 3
J: 4, 6, 7
K: 3, 7, 8
PS6: (9) Due 3/25
1C.1: 1, 2, 6
1C.2: 6, 11, 17
OR PS1 from UW Math 301
PS7: (8) Due 4/1
1D.1: 1, 13, 14, 16
1D.2: 2, 3
1D.3: 1abc, 4
OR PS2 from UW Math 301
PS8: (10) Due 4/15
1E: 6, 11
1F.1: 9, 10, 12, 21
1F.2: 1be, 2bg
1G: 3, 6
OR PS3 from UW Math 301
PS9: (8) Due 4/22
1H: 8, 16, 17
1I: 1, 2
1J: 9, 12, 13
OR PS4 from UW Math 301
PS10: (5) Due 4/29
2A: 7, 9, 15
2B: 2, 6
PS11: (11) Due 5/6
2C.2: 5, 7, 11, 12
2C.3: 2, 4, 7, 8, 9
PS12: (8) Due 5/13
2D.1: 2, 3
2D.2: 1-5, 9
PS13: (6) Due 5/20
2E.1: 2, 7
2E.3: 1, 3
PS14: (4) Due 5/27
2F: 2, 5
2G: 2, 5
A.1 Elements (or members), well-defined, Cardinality n(A), Set-Builder Notation, Equality of Sets, Empty Set and Universal Set, Optional: Infinite Cardinalities
A.2 Subsets, Trivial Subsets, Proper Subsets, Venn Diagrams, Power set,
A.3 Operations on Sets: Intersection, Union, and Complement
A.4 Associativity and Distributivity, DeMorgan’s Laws
A.5 Difference and Symmetric Difference of Sets. Commutativity and Associativity of Symmetric Difference
B. Ordered Pairs
B.1 Cartesian Product
B.2 Relations, Domain and Range, Reflexive Relations, Symmetric Relations, Transitive Relations
B.3 Equivalence Relations, Empty Relation, Thm 1: Equivalence Relations partition a set into disjoint equivalence classes.
B.4 Residue Classes modulo n, Congruence Modulo n, Thm 2: A Bunch of Equivalent Statements are Equivalent
C.1 Functions as a type of relation, functions as mappings, domain, codomain, and range, injections, surjections, bijections
C.2 Composition, Inverse Functions,
D. Binary Operations
D.1 Closure under an operation
D.2 Associativity, Using Exponent Notation, Commutativity, Distributivity
D.3 Thm 3: Uniqueness of Identity, Thm 4: Uniqueness of Inverses
D.4 Cayley Tables (aka multiplication tables)
E.1 Definition of Group, Thm 5: Cancellation in Groups is OK, Thm 6: The Cayley Table for a Group is a Sudoku board, Abelian Groups
E.2 Order of a Group, Order of an Element, Thm 7: Elements of Finite Groups have Finite Order, Thm 8: Inverse of inverse cancels out and inverse of a^n
Investigation: Groups of Small Order
F. Permutation Groups
F.1 Permutation of sets. Composing permutations, Identity Permutation, Inverse Permutations. Thm 9: S_n is a group.
F.2 Cycle Notation. Composition using Cycle Notation. Thm 10: Any permutation can be written as a composition of disjoint cycles. Thm 11: Disjoint cycles commute. Thm 12: Order of Cycles and Inverse of Cycles. Thm 13: Order of a composition of cycles is the LCM of the orders of the cycles.
F.3 Symmetries of Plane Figures aka Dihedral Groups
Trivial Subgroup. Proper Subgroups. Nested Subgroups. Subgroup as a Relation. Thm 14: The subset generated by a single element forms a subgroup. Thm 15: Subgroup Test. Thm 16: Subgroup Test for Finite Groups.
H. Cyclic Groups
Generators, Infinite Cyclic Group, Thm 17: All Cyclic Groups are Abelian. Thm 18: A finite cyclic group of order n has a generator g or order n and the group consists of g, g^2, …. Thm 19: For any integer n, there exists a cyclic group of order n. Galois. Thm 20: Subgroups of cyclic groups are cyclic.
Definition of homomorphism. Kernel. Range. Isomorphism. Isomorphic. Thm 21: Homomorphisms respect identity and respect inverses. Thm 22: A Homomorphism is an injection if and only if the Kernel consists of only the identity. Thm 23: The Kernel and Range form subgroups of their respective parent groups.
Properties of Isomorphisms. Thm 24: All cyclic groups of order n are isomorphic to each other.
K. Cosets and LaGrange’s Theorem
Left cosets and right cosets. Thm 25. Left and Right cosets of abelian groups are equal.
Appendix: Methods of Proof