Math 215 Introduction to Advanced Mathematics
Fall 2013
- Week 1 basics proofs using absolute values, proof by cases,
mathematical statements, truth tables for basic connectives
Read Chapters 1 and 2
- Week 2 converses and contrapositives, direct proof, axioms for groups and
sample proofs about groups
Read Chapters 3
- Week 3
Proofs by cases, proof of contrapositive, proof by contradiction.
proof of irrationality of square root of 2, proof there are infinitely many prime numbers
Read Chapters 4
- Week 4 proofs by induction, strong induction, Fibonacci numbers
Read Chapter 5
- Week 5 sets, subsets, equality of sets, operations on sets: union, intersection, difference, complements
Read Chapter 6
- Week 6 power set, cartesian products, quantifiers, alternation of quantifiers,
Midterm 1
- Week 7 quantifiers, alternation of quantifiers, negating quantifier strings,
convergence of sequences.
Read Chapter 7
- Week 8 functions, graphs, injections, surjections, bijections
Read Chapter 8, Chapter 9
- Week 9 inverse functions, images and inverse images, equivalence relations
Read Chapter 22
- Week 10 partitions, counting finite sets, the addition principle, the inclusion-exclusion principle
Read Chapter 10, 11
- Week 11 the pigeonhole principle, Dirichlet's Diophantine Approximation Theorem, subsets of finite sets are finite, counting sets of functions,
Read Chapter 12
- Week 12counting injections, counting subsets, binomial coefficients, the binomial theorem, Midterm II
- Week 13 the binomial theorem, counting infinite sets, denumerable sets, countability of the rationals, Cantor's Theorem on the uncountability of the reals
Read Chapter 14
- Week 14 countable unions of countable sets are countable, existence of transcendental numbers, the division theorem, the Euclidean Algorithm
Read Chapter 15, 16
- Week 15 writing the gcd as an integer linear combination, solving linear diophantine equations, review
Read Chapter 17.1, 18
Last revised: 11/11/13