Title
Preface
Contents
Symbols used and pages where ๏ฌrst introduced
1. Sets
1. De๏ฌnition
2. Set Membership and Set Inclusion
3. Union and Intersection of Sets
4. Ordered Pairs
5. Binary Operations
2. The Natural Numbers
1. The Problem
2. Closure
3. The Commutative and Associative Laws
4. The Distributive Law for Multiplication with Respect to Addition
5. The Remaining Postulates
6. Isomorphism
7. The Principle of Finite Induction
8. The Peano Postulates
3. Equivalent Pairs of Natural Numbers
1. Equivalence
2. De๏ฌnition of Addition and Multiplication
3. Properties of Addition and Multiplication
4. Equivalence Classes and The Integers
1. Relations
2. Partitioning of a Set
3. Constructing the Integers
4. Further Properties of the Integers
5. The Natural Numbers as a Subset of the Integers
6. Zero and Subtraction
5. Integral Domains
1. De๏ฌnition of an Integral Domain
2. Elementary Properties of Integral Domains
3. Division in an Integral Domain
4. Ordered Integral Domains
5. Variations of the Principle of Mathematical Induction
6. The Greatest Common Divisor
7. Unique Factorization in I
6. The Rational Numbers
1. Equivalence
2. Addition and Multiplication of Rational Numbers
3. Properties of the Rational Numbers
4. The Integers as a Subset of the Rational Numbers
5. Additive and Multiplicative Inverses in R
6. The Ordering of the Rational Numbers
7. Groups and Fields
1. De๏ฌnitions and Examples
2. Further Examples of Groups
3. Some Simple Properties of Groups
4. Permutations
5. Mappings
6. Isomorphisms and Automorphisms of Groups
7. Automorphisms of a Field
8. Cyclic Groups
9. Subgroups
8. The Real Numbers
1. Rational Numbers as Decimals
2. Sequences
3. Cauchy Sequences
4. Null Sequences
5. The Real Numbers
6. The Rational Numbers as Real Numbers
7. sqrt(2) as a Real Number
8. Ordering of the Real Numbers
9. The Field of Real Numbers
10. Additional Remarks
9. Rings, Ideals, and Homomorphisms
1. Subrings and Ideals
2. Residue Class Rings
3. Homomorphisms
4. Homomorphisms of Groups
5. Lagrangeโs Theorem
10. Complex Numbers and Quaternions
1. Complex Numbers as Residue Classes
2. Complex Numbers as Pairs of Real Numbers
3. Algebraically Closed Fields
4. De Moivreโs Theorem
5. Quaternions
11. Vector Spaces
1. De๏ฌnition
2. The Basis of a Vector Space
3. The Dimension of a Vector Space
4. Fields as Vector Spaces
12. Polynomials
1. De๏ฌnitions
2. The Division Algorithm
3. Greatest Common Divisor
4. Zeros of a Polynomial
5. The Cubic and Quartic Equations
Bibliography
Index