x Contents 1.4 Relations 32 What Is a Relation? 32 Functions as Relations 33 Properties of Relations 33 Equivalence Relations 36 Partial and Total Orders 39 Important Concepts,Formulas,and Theorems 41 Problems 42 1.5 Using Equivalence Relations in Counting 43 The Symmetry Principle 43 Equivalence Relations 45 The Quotient Principle 46 Equivalence Class Counting 46 Multisets 48 The Bookcase Arrangement Problem 50 The Number of k-Element Multisets of an n-Element Set 51 Using the Quotient Principle to Explain a Quotient 52 Important Concepts,Formulas,and Theorems 53 Problems 54 CHAPTER 2 Cryptography and Number Theory 59 2.1 Cryptography and Modular Arithmetic 59 Introduction to Cryptography 59 Private-Key Cryptography 60 Public-Key Cryptosystems 63 Arithmetic Modulo n 65 Cryptography Using Addition mod n 68 Cryptography Using Multiplication mod n 69 Important Concepts,Formulas,and Theorems 71 Problems 72x Contents 1.4 Relations 32 What Is a Relation? 32 Functions as Relations 33 Properties of Relations 33 Equivalence Relations 36 Partial and Total Orders 39 Important Concepts, Formulas, and Theorems 41 Problems 42 1.5 Using Equivalence Relations in Counting 43 The Symmetry Principle 43 Equivalence Relations 45 The Quotient Principle 46 Equivalence Class Counting 46 Multisets 48 The Bookcase Arrangement Problem 50 The Number of k-Element Multisets of an n-Element Set 51 Using the Quotient Principle to Explain a Quotient 52 Important Concepts, Formulas, and Theorems 53 Problems 54 CHAPTER 2 Cryptography and Number Theory 59 2.1 Cryptography and Modular Arithmetic 59 Introduction to Cryptography 59 Private-Key Cryptography 60 Public-Key Cryptosystems 63 Arithmetic Modulo n 65 Cryptography Using Addition mod n 68 Cryptography Using Multiplication mod n 69 Important Concepts, Formulas, and Theorems 71 Problems 72