This text is intended to help students move from introductory courses to those where rigor and proof play a much greater role. The emphasis is on precise definitions of mathematical objects and rigorous proofs of properties. The text contains four parts. The first reviews earlier concepts more rigorously and covers methods to make a correct argument. In the second part, concepts and objects are introduced including induction, sets, functions, cardinality, complex numbers, permutations, and matrices. The third part covers number theory including applications to cryptography. In the final part, important objects from abstract algebra are introduced at a relatively elementary level.
1 A Look Back: Precalculus Math 2 A Look Back: Calculus 3 About Proofs and Proof Strategies 4 Mathematical Induction 5 The Well-ordering Principle 6 Sets 7 Equivalence Relations 8 Functions 9 Cardinality of Sets 10 Permutations 11 Complex Numbers 12 Matrices and Sets with Algebraic Structure 13 Divisibility in Z and Number Theory 14 Primes and Unique Factorization 15 Congruences and the Finite Sets Zn 16 Solving Congruences 17 Fermat’s Theorem 18 Diffie-Hellman Key Exchange 19 Euler’s Function and Euler’s Theorem