Introduction to Abstract Mathematics

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2. The Principle of Induction......Page 5 3. The Principle of Strong Induction......Page 10 4. The Binomial Theorem......Page 13 2. The Division Algorithm......Page 18 3. Euclid's Algorithm......Page 20 4. The equation ax+by=gcd(a,b)......Page 21 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 32 5. Image and preimage......Page 35 1. Equivalence relations and partitions......Page 38 2. The Chinese Remainder Theorem......Page 43 3. Arithmetic in Z/nZ......Page 46 4. Rings and the units of Z/nZ......Page 52 5. Fermat's Little Theorem......Page 55 1. Integral domains and fields......Page 58 2. The division algorithm for polynomials......Page 60 3. Euclid's Algorithm for polynomials......Page 61 4. Unique factorization of polynomials......Page 64 5. Roots of polynomials......Page 67 6. Eisenstein's criterion and the Gauss lemma......Page 70