Introduction to Abstract Mathematics

1. The irrationality of 2......Page 5 2. The Principle of Induction......Page 6 3. The Principle of Strong Induction......Page 11 4. The Binomial Theorem......Page 14 1. Divisibility and the GCD......Page 17 2. The Division Algorithm......Page 18 3. Euclid's Algorithm......Page 19 4. The equation ax+by=gcd(a,b)......Page 20 5. The Fundamental Theorem of Arithmetic......Page 24 1. The complex numbers......Page 26 2. Roots of unity......Page 27 3. Operations on sets......Page 30 4. Functions......Page 31 5. Image and preimage......Page 36 1. Equivalence relations and partitions......Page 38 2. The Chinese Remainder Theorem......Page 43 3. Arithmetic in Z/nZ......Page 47 4. Rings and the units of Z/nZ......Page 54 5. Fermat's Little Theorem......Page 57 6. Euler's function......Page 60 7. Euler's theorem......Page 63 1. Integral domains and fields......Page 65 2. The division algorithm for polynomials......Page 68 3. Euclid's Algorithm for polynomials......Page 70 4. Unique factorization of polynomials......Page 73 5. Roots of polynomials......Page 76 7. Eisenstein's criterion and the Gauss lemma......Page 81