PACKAGING & DIELINES - THE DESIGNER’S BOOK OF PACKAGING DIELINES

Emphasizing the creative nature of mathematics, this conversational textbook guides students through the process of discovering a proof. The material revolves around possible strategies to approaching a problem without classifying 'types of proofs' or providing proof templates. Instead, it helps students develop the thinking skills needed to tackle mathematics when there is no clear algorithm or recipe to follow. Beginning by discussing familiar and fundamental topics from a more theoretical perspective, the book moves on to inequalities, induction, relations, cardinality, and elementary number theory. The final supplementary chapters allow students to apply these strategies to the topics they will learn in future courses. With its focus on 'doing mathematics' through 200 worked examples, over 370 problems, illustrations, discussions, and minimal prerequisites, this course will be indispensable to first- and second-year students in mathematics, statistics, and computer science. Instructor resources include solutions to select problems. Cover Half-title page Series page Title page Copyright page Reviews Contents List of Symbols and Notation Preface Part I Core Material 1 Numbers, Quadratics and Inequalities 1.1 The Quadratic Formula 1.2 Working with Inequalities – Setting the Stage 1.3 The Arithmetic-Geometric Mean and the Triangle Inequalities 1.4 Types of Numbers 1.5 Problems 1.6 Solutions to Exercises 2 Sets, Functions and the Field Axioms 2.1 Sets Notation and Terminology The Interval Notation and Set Operations 2.2 Functions 2.3 The Field Axioms The Real Numbers System 2.4 Appendix: Infinite Unions and Intersections 2.5 Appendix: Defining Functions 2.6 Problems 2.7 Solutions to Exercises 3 Informal Logic and Proof Strategies 3.1 Mathematical Statements and their Building Blocks Quantifiers Connectives 3.2 The Logic Symbols 3.3 Truth and Falsity of Compound Statements Implications A Remark on Quantifiers 3.4 Truth Tables and Logical Equivalences Equivalences Involving Quantifiers 3.5 Negation 3.6 Proof Strategies Direct Proof Proof by Contrapositive Proof by Contradiction 3.7 Problems 3.8 Solutions to Exercises 4 Mathematical Induction 4.1 The Principle of Mathematical Induction 4.2 Summation and Product Notation 4.3 Variations Variation 1 Variation 2 Variation 3 4.4 Additional Examples Recursive Definitions of Sequences A Fallacy 4.5 Strong Mathematical Induction The Principle of Strong Mathematical Induction Proof of Existence Proof of Uniqueness 4.6 Problems 4.7 Solutions to Exercises 5 Bijections and Cardinality 5.1 Injections, Surjections and Bijections 5.2 Compositions 5.3 Cardinality 5.4 Cardinality Theorems 5.5 More Cardinality and the Schröder–Bernstein Theorem 5.6 Problems 5.7 Solutions to Exercises 6 Integers and Divisibility 6.1 Divisibility and the Division Algorithm Proof of Uniqueness 6.2 Greatest Common Divisors and the Euclidean Algorithm The Euclidean Algorithm 6.3 The Fundamental Theorem of Arithmetic 6.4 Problems 6.5 Solutions to Exercises 7 Relations 7.1 The Definition of a Relation Symbols for Commonly Used Relations 7.2 Equivalence Relations 7.3 Equivalence Classes 7.4 Congruence Modulo n 7.5 Problems 7.6 Solutions to Exercises Part II Additional Topics 8 Elementary Combinatorics 8.1 Counting Arguments: Selections, Arrangements and Permutations 8.2 The Binomial Theorem and Pascal’s Triangle 8.3 The Pigeonhole Principle 8.4 The Inclusion-Exclusion Principle 8.5 Problems 8.6 Solutions to Exercises 9 Preview of Real Analysis – Limits and Continuity 9.1 The Limit of a Sequence 9.2 The Limit of a Function One-Sided Limits 9.3 The Relation between Limits of Functions and Sequences 9.4 Continuity and Differentiability Continuity Differentiability 9.5 Problems 9.6 Solutions to Exercises 10 Complex Numbers 10.1 Background 10.2 The Field of Complex Numbers 10.3 The Complex Plane and the Triangle Inequality Sums of Complex Numbers The Triangle Inequality 10.4 Square Roots and Quadratic Equations Complex Quadratic Equations 10.5 Polar Representation of Complex Numbers Geometric Interpretation of Products 10.6 De Moivre’s Theorem and Roots Arbitrary Roots of Complex Numbers General Polynomial Equations 10.7 The Exponential Function 10.8 Problems 10.9 Solutions to Exercises 11 Preview of Linear Algebra 11.1 The Spaces R[sup(n)] and their Properties 11.2 Geometric Vectors Scalar Multiplication 11.3 Abstract Vector Spaces 11.4 Subspaces 11.5 Linear Maps and Isomorphisms 11.6 Problems 11.7 Solutions to Exercises Index Emphasizing the creative nature of mathematics, this conversational textbook guides students through the process of discovering a proof as they transition to advanced mathematics. Using several strategies, students will develop the thinking skills needed to tackle mathematics when there is no clear algorithm or recipe to follow.