Preface ix ■S anded coverage ofalgorithms(Chapter3)and number More second and third level heads have beenused to break sections into smaller coherent parts Tools for Easier Learning Difficult discussions and proofs have been marked with the famous Bourbaki"dangerous bend"symbol in the margin. New marginal notes make connections,add interesting notes,and provide advice to students. added explanations,in both proofs and exposition,make iteasier for Many new exercises,both routine and challenging,have been added,while many ex- isting exercises have been improved. Enhanced Coverage of Logic,Sets,and Proof The satisfiability problem is addressed in greater depth,with Sudoku modeled in terms of satisfiability Hilbert's Grand Hotel is used to help explain uncountability Proofs throughout the book have been made more accessible by adding steps and reasons behind these steps. A template for proofs by mathematical induction has been added The step that applies the inductive hypothesis in mathematical induction proof is now explicitly noted. Algorithms The pseudocode used in the book has been updated. Useful rules for big-O estimates of logarithms,powers,and exponential functions have been added. Number Theory and Cryptography eltionhipbetween the mod function and congrunceshas been explained more The sieve of Eratosthenes is now introduced earlier in the book Linear congruences and modular inverses are now covered in more detail Applications of number theory,including check digits and hash functions,are covered in great depth. ■A new section o hy integrates previous the notion Cryptographic protocols,including digital signatures and key sharing,are now coveredPreface ix Separate chapters now provide expanded coverage of algorithms (Chapter 3) and number theory and cryptography (Chapter 4). More second and third level heads have been used to break sections into smaller coherent parts. Tools for Easier Learning Difficult discussions and proofs have been marked with the famous Bourbaki “dangerous bend” symbol in the margin. New marginal notes make connections, add interesting notes, and provide advice to students. More details and added explanations, in both proofs and exposition, make it easier for students to read the book. Many new exercises, both routine and challenging, have been added, while many ex￾isting exercises have been improved. Enhanced Coverage of Logic, Sets, and Proof The satisfiability problem is addressed in greater depth, with Sudoku modeled in terms of satisfiability. Hilbert’s Grand Hotel is used to help explain uncountability. Proofs throughout the book have been made more accessible by adding steps and reasons behind these steps. A template for proofs by mathematical induction has been added. The step that applies the inductive hypothesis in mathematical induction proof is now explicitly noted. Algorithms The pseudocode used in the book has been updated. Explicit coverage of algorithmic paradigms, including brute force, greedy algorithms, and dynamic programing, is now provided. Useful rules for big-O estimates of logarithms, powers, and exponential functions have been added. Number Theory and Cryptography Expanded coverage allows instructors to include just a little or a lot of number theory in their courses. The relationship between the mod function and congruences has been explained more fully. The sieve of Eratosthenes is now introduced earlier in the book. Linear congruences and modular inverses are now covered in more detail. Applications of number theory, including check digits and hash functions, are covered in great depth. A new section on cryptography integrates previous coverage, and the notion of a cryp￾tosystem has been introduced. Cryptographic protocols, including digital signatures and key sharing, are now covered.