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Discrete Mathematics with Applications


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1. THE LOGIC OF COMPOUND STATEMENTS. Logical Form and Logical Equivalence. Conditional Statements. Valid and Invalid Arguments. Application: Digital Logic Circuits. Application: Number Systems and Circuits for Addition. 2. THE LOGIC OF QUANTIFIED STATEMENTS. Introduction to Predicates and Quantified Statements I. Introduction to Predicates and Quantified Statements II. Statements Containing Multiple Quantifiers. Arguments with Quantified Statements. 3. ELEMENTARY NUMBER THEORY AND METHODS OF PROOF. Direct Proof and Counterexample I: Introduction. Direct Proof and Counterexample II: Rational Numbers. Direct Proof and Counterexample III: Divisibility. Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem. Direct Proof and Counterexample V: Floor and Ceiling. Indirect Argument: Contradiction and Contraposition. Two Classical Theorems. Application: Algorithms. 4. SEQUENCES AND MATHEMATICAL INDUCTION. Sequences. Mathematical Induction I. Mathematical Induction II. Strong Mathematical Induction and the Well-Ordering Principle. Application: Correctness of Algorithms. 5. SET THEORY. Basic Definitions of Set Theory. Properties of Sets. Disproofs, Algebraic Proofs, and Boolean Algebras. Russell's Paradox and the Halting Problem. 6. COUNTING AND PROBABILITY. Introduction. Possibility Trees and the Multiplication Rule. Counting Elements of Disjoint Sets: The Addition Rule. Counting Subsets of a Set: Combinations. R-Combinations with Repetition Allowed. The Algebra of Combinations. The Binomial Theorem. Probability Axioms and Expected Value. Conditional Probability, Bayes' Formula, and Independent Events. 7. FUNCTIONS. Functions Defined on General Sets. One-to-One and Onto, Inverse Functions. Application: The Pigeonhole Principle. Composition of Functions. Cardinality with Applications to Computability. 8. RECURSION. Recursively Defined Sequences. Solving Recurrence Relations by Iteration. Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients. General Recursive Definitions. 9. THE EFFICIENCY OF ALGORITHMS. Real-Valued Functions of a Real Variable and Their Graphs. O-, Omega-, and Theta-Notations. Application: Efficiency of Algorithms I. Exponential and Logarithmic Functions: Graphs and Orders. Application: Efficiency of Algorithms II. 10. RELATIONS. Relations on Sets. Reflexivity, Symmetry, and Transitivity. Equivalence Relations. Modular Arithmetic with Applications to Cryptography. Partial Order Relations. 11. GRAPHS AND TREES. Graphs: An Introduction. Paths and Circuits. Matrix Representations of Graphs. Isomorphisms of Graphs. Trees. Spanning Trees. 12. FINITE STATE AUTOMATA AND APPLICATIONS. Finite-State Automata. Application: Regular Expressions. Finite-State Automata. Simplifying Finite-State Automata. Appendices. Properties of the Real Numbers. Solutions and Hints to Selected Exercises.

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