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Full Description
Passage to Abstract Mathematics facilitates the transition from introductory mathematics courses to the more abstract work that occurs in advanced courses. This text covers logic, proofs, numbers, sets, induction, functions, and more-material which instructors of upper-level courses often presume their students have already mastered but are in fact missing from lower-level courses. Students will learn how to read and write mathematics-especially proofs-the way that mathematicians do. The text emphasizes the use of complete, correct definitions and mathematical syntax.
Contents
Preface for the Instructor viiPreface for the Student xiii1. Logic and Proof 1.1 Proofs, what and why?1.2 Statements and Non-statements1.3 Logical Operations and Logical Equivalence1.4 Conditionals, Tautologies and Contradictions1.5 Methods of Proof1.6 Quantifiers1.7 Further Exercises2. Numbers 2.1 Basic Ideas of Sets2.2 Sets of Numbers2.3 Some properties of N and Z2.4 Prime Numbers2.5 gcd's and lcm's2.6 Euclid's Algorithm2.7 Rational Numbers and Algebraic Numbers2.8 Further Exercises3. Sets 3.1 Subsets3.2 Operations with Sets3.3 The Complement of a Set3.4 The Cartesian Product3.5 Families of Sets3.6 Further Exercises4. Induction 4.1 An Inductive Example4.2 The Principle of Mathematical Induction4.3 The Principle of Strong Induction4.4 The Binomial Theorem4.5 Further Exercises5. Functions5.1 Functional Notation5.2 Operations on Functions5.3 Induced Set Functions5.4 Surjections, Injections, and Bijections5.5 Identity Functions, Cancellation, Inverse Functions, and Restrictions5.6 Further Exercises6. Binary Relations6.1 Partitions6.2 Equivalence Relations6.3 Order Relations6.4 Bounds and Extremal Elements6.5 Applications to Calculus6.6 Functions Revisited6.7 Further Exercises7. Infinite Sets and Cardinality 7.1 Counting7.2 Properties of Countable Sets7.3 Counting Countable Sets7.4 Binary Relations on Cardinal Numbers7.5 Uncountable Sets7.6 Further Exercises8. Algebraic Systems 8.1 Binary Operations8.2 Modular Arithmetic8.3 Numbers Revisited8.4 Complex Numbers8.5 Further ExercisesIndex of Symbols and NotationIndex
