# 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

• Week 2 converses and contrapositives, direct proof, axioms for groups and sample proofs about groups

• 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

• Week 4 proofs by induction, strong induction, Fibonacci numbers

• Week 5 sets, subsets, equality of sets, operations on sets: union, intersection, difference, complements

• Week 6 power set, cartesian products, quantifiers, alternation of quantifiers, Midterm 1

• Week 7 quantifiers, alternation of quantifiers, negating quantifier strings, convergence of sequences.

• Week 8 functions, graphs, injections, surjections, bijections

• Week 9 inverse functions, images and inverse images, equivalence relations

• Week 10 partitions, counting finite sets, the addition principle, the inclusion-exclusion principle

• Week 11 the pigeonhole principle, Dirichlet's Diophantine Approximation Theorem, subsets of finite sets are finite, counting sets of functions,

• 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