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| preface 1 Truth Tables,Logic,and Proofs 1.1 Statements and Connectives 1.2 Conditional Statements 1.3 Equivalent Statements 1.4 Axiomatic Systems:Arguments and Proofs 1.5 Completeness in Propositional Logic 1.6 Karnaugh Maps 1.7 Circuit Diagrams 2 Set Theory 2.1 Introduction to Sets 2.2 Set Operations 2.3 Venn Diagrams 2.4 Boolean Algebras 2.5 Relations 2.6 Partially Ordered Sets 2.7 Equivalence Relations 3 Logic,Integers,and Proofs 3.1 Predicate Calculus 3.2 Basic Concepts of Proofs and the Structure of Integers 3.3 Mathematical Induction 3.4 Divisibility 3.5 Prime Integers 3.6 Congruence Relations 4 Functions and Matrices 4.1 Functions 4.2 Special Functions 4.3 Matrices 4.4 Cardinality 4.5 Cardinals Continued 5 Algorithms and Recursion 5.1 The “for” Procedure and Algorithms for Matrices 5.2 Recursive Functions and Algorithms 5.3 Complexity of Algorithms 5.4 Sorting Algorithms 5.5 Prefix and Suffix Notation 5.6 Binary and Hexadecimal Numbers 5.7 Signed Numbers 5.8 Matrices Continued 6 Graphs,Directed Graphs,and Trees 6.1 Graphs 6.2 Directed Graphs 6.3 Trees 6.4 Instant Insanity 6.5 Euler Paths and Cycles 6.6 Incidence and Adjacency Matrices 6.7 Hypercubes and Gray Code 7 Number Theory 7.1 Sieve of Eratosthenes 7.2 Fermat's Factorization Method 7.3 The Division and Euclidean Algorithms 7.4 Continued Fractions 7.5 Convergents 8 Counting and Probability 8.1 Basic Counting Principles 8.2 Inclusion-Exclusion Introduced 8.3 Permutations and Combinations 8.4 Generating Permutations and Combinations 8.5 Probability Introduced 8.6 Generalized Permutations and Combinations 8.7 Permutations and Combinations with Repeition 8.8 Pigeonhole Principle 8.9 Probability Revisited 8.10 Bayes' Theorem 8.11 Markov Chains 9 Algebraic Structures 10 Number Theory Revisited 11 Recursion Revisited 12 Counting Continued 13 Generating Functions 14 Graphs Revisited 15 Trees 16 Networks 17 Theory of Computation 18 Theory of Codes 19 Enumeration of Colors 20 Rings,Integral Domains,and Fields 21 Group and Semigroup Characters 22 Applications of Number Theory Bibliography Hints and Solutions to Selected Exercises Index |
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