Math 215 Introduction to Advanced Mathematics
Fall 2012
- Week 1 mathematical statements, logical connectives,
truth tables, negation, implications
Read Chapters 1 and 2
- Week 2 Proofs, direct proofs, proofs by cases, axioms for groups, axioms for
ordered fields
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, power set, operations on sets: union, intersection, difference, complements \
Read Chapter 6
- Week 6 cartesian products, quantifiers, alternation of quantifiers,
sequences
Read Chapter 7
- Week 7 convergence of sequences, functions, graphs,
Read Chapter 8
Midterm I Friday October 12
- Week 8 injections, surjections, bijections, inverse functions,
Read Chapter 9
- Week 9 images and inverse images, partitions, equivalence relations
Read Chapter 22
- Week 10 counting finite sets, the addition principle, the inclusion-exclusion principle, the pigeonhole principle, Dirichlet's Diophantine Approximation Theorem
Read Chapter 10, 11
- Week 11 subsets of finite sets are finite, counting sets of functions, counting injections, counting subsets
Read Chapter 12
- Week 12 binomial coefficients, the binomial theorem, Midterm II
- Week 13 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: 11/28/12